Mathematical Congress of the Americas

Home , Dennis DeTurck, Hopf conjecture, Lawrence Zalcman, Yuri Burago

656

First Mathematical Congress of the Americas August 5–9, 2013 Guanajuato, México

José A. de la Peña J. Alfredo López-Mimbela Miguel Nakamura Jimmy Petean Editors

American Mathematical Society

Mathematical Congress of the Americas

First Mathematical Congress of the Americas August 5–9, 2013 Guanajuato, México

José A. de la Peña J. Alfredo López-Mimbela Miguel Nakamura Jimmy Petean Editors

656

Mathematical Congress of the Americas

First Mathematical Congress of the Americas August 5–9, 2013 Guanajuato, México

José A. de la Peña J. Alfredo López-Mimbela Miguel Nakamura Jimmy Petean Editors

American Mathematical Society Providence, Rhode Island

EDITORIAL COMMITTEE Dennis DeTurck, Managing Editor Michael Loss Kailash Misra Martin J. Strauss

2000 Mathematics Subject Classiﬁcation. Primary 00-02, 00A05, 00A99, 00B20, 00B25.

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Contemporary Mathematics ISSN: 0271-4132 (print); ISSN: 1098-3627 (online)

DOI: http://dx.doi.org/10.1090/conm/656

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Contents

Preface vii Symmetries, Hopf fibrations and supercritical elliptic problems Monica´ Clapp and Angela Pistoia 1 Min-max theory of minimal surfaces and applications Fernando C. Marques and Andre´ Neves 13 Homogenization on manifolds Gonzalo Contreras 27 Lagrangian cobordism: Rigidity and flexibility aspects Octav Cornea 41 Biochemical reaction networks: An invitation for algebraic geometers Alicia Dickenstein 65 Long-time asymptotic expansions for nonlinear diffusions in Euclidean space Jochen Denzler, Herbert Koch, and Robert J. McCann 85 Non-strongly isospectral spherical space forms E.A.Lauret,R.J.Miatello,and J. P. Rossetti 95 Entrance laws for positive self-similar Markov processes V´ıctor Rivero 119 Combinatorics and geometry Fernando Rodriguez-Villegas 141 A (short) survey on dominated splittings M. Sambarino 149 Geometric regularity estimates for elliptic equations Eduardo V. Teixeira 185

Preface

In January 2011, during the AMS meeting in New Orleans, representatives of the major mathematical societies of the continent agreed to create the Mathematical Congress of the Americas. Months later, a meeting ”Mathematics in the Americas” was held at IMPA, Rio de Janeiro, where it was decided that the inaugural MCA would take place in Guanajuato, Mexico, on August 5–9, 2013. With a four-year periodicity, the goal of the Congress is to highlight the excellence of mathematical achievements in the Americas within the context of the international arena and to foster the scientific integration of all mathematical communities in the continent. Guanajuato is a historic city designated by UNESCO as World Heritage. The selection of this city to host such an important meeting was proposed by CIMAT, the Center of Mathematics of Guanajuato, one of the important centers of research in México. The response to the call for participation was excellent. Essential for the response of the academic community to the MCA2013 was the Steering Committee of the Congress. The international leadership and ad- vice of Susan Friedlander (AMS), Marcelo Viana (SMB), Alejandro Adem (CMS), Servet Mart´ınez (UMALCA) and Uri Ascher (SIAM) were always ready and important. They, plus JoséA.delaPeña (SMM), represented the sponsor organizations of the MCA2013: the American Mathematical Society, the Sociedad Matematica Brasileira, the Canadian Mathematical Society, the Unión Matemática de América Latina y el Caribe, the Society for Industrial and Applied Mathematics and the So- ciedad Matemática Mexicana. The work of many other mathematicians should be acknowledged: the program Committee, the Prize Committee, the local organizing Committee formed by colleagues of CIMAT and the state University of Guanaju- ato, and many others who helped to organize the participation of close to 1,000 researchers and students from more than 40 countries of the continent and beyond. The MCA2013 defined, no doubt, a benchmark for mathematics in the continent. The Program Committee contributed to this purpose by selecting an ex- ceptional group of distinguished mathematicians as plenary and invited speakers of the meeting. Those mathematicians, as well as the winners of MCA awards, were invited to submit papers to this volume. In this way, the Proceedings of the First Mathematical Congress of the Americas is a small testimony of the state of the art of mathematics in the Americas. It is a pleasure that the American Mathematical Society accepted to publish the Proceedings in their Contemporary Mathematics series. Last, but not least, the financial support of Consejo Nacional de Ciencia y Tec- nolog´ıa, México was fundamental for the success of the Congress. The local support of CIMAT was instrumental for the smooth running of every aspect (and there were many!) of the event: starting the preparations two years before MCA2013, during

vii

viii PREFACE the Congress and afterwards, closing the work with the edition of these Proceed- ings. It is a pleasure to thank all institutions and people whose commitment and work made possible the First Mathematical Congress of the Americas. The Editorial Committee José-AntoniodelaPeña JoséAlfredoLópez-Mimbela Miguel Nakamura Jimmy Petean Centro de Investigación en Matemáticas, Guanajuato, México.

Abril, 2015.

Contemporary Mathematics Volume 656, 2016 http://dx.doi.org/10.1090/conm/656/13100

Symmetries, Hopf ﬁbrations and supercritical elliptic problems

M´onica Clapp and Angela Pistoia

Abstract. We consider the semilinear elliptic boundary value problem − −Δu = |u|p 2 u in Ω,u=0on∂Ω, RN 2N in a bounded smooth domain Ω of for supercritical exponents p> N−2 . Until recently, only few existence results were known. An approach which has been successfully applied to study this problem, consists in reducing it to a more general critical or subcritical problem, either by considering rotational symmetries, or by means of maps which preserve the Laplace operator, or by a combination of both. The aim of this paper is to illustrate this approach by presenting a selection of recent results where it is used to establish existence and multiplicity or to study the concentration behavior of solutions at supercritical exponents.

1. Introduction Consider the model problem − −Δu = |u|p 2 u in Ω, (℘ ) p u =0 on∂Ω, where Δ is the Laplace operator, Ω is a bounded domain in RN with smooth boundary, N ≥ 3, and p>2. Despite its simple form, this problem has been an amazing source of open problems, and the process of understanding it has helped develop new and interesting techniques which can be applied to a wide variety of problems. The behavior of this problem depends strongly on the exponent p. It is called ∈ ∗ ∗ subcritical, critical or supercritical depending on whether p (2, 2N ),p=2N or ∈ ∗ ∞ ∗ 2N p (2N , ), where 2N := N−2 is the so-called critical Sobolev exponent. In the subcritical case, standard variational methods yield the existence of a positive solution and inﬁnitely many sign changing solutions. But if p is critical or supercritical the existence of solutions becomes a delicate issue. It depends on the domain. An identity obtained by Pohozhaev [27]impliesthat(℘p)doesnot

2010 Mathematics Subject Classification. Primary 35J61; Secondary 35J20, 35J25. Key words and phrases. Nonlinear elliptic boundary value problem, supercritical nonlinearity, nonautonomous critical problem. Research supported by CONACYT grant 129847 and PAPIIT grant IN106612 (Mexico) and Università degli Studi di Roma ”La Sapienza” Accordi Bilaterali ”Esistenza e proprietàgeomet- riche di soluzioni di equazioni ellittiche non lineari” (Italy).

c 2016 American Mathematical Society 1

2MONICA´ CLAPP AND ANGELA PISTOIA

∈ ∗ ∞ have a nontrivial solution if Ω is strictly starshaped and p [2N , ). On the other hand, Kazdan and Warner [18] showed that infinitely many radial solutions exist for every p ∈ (2, ∞) if Ω is an annulus. The critical problem has received much attention during the last thirty years, partly due to the fact that it is a simple model for equations which arise in some fundamental questions in differential geometry, like the Yamabe problem or the prescribed scalar curvature problem. Still, many questions remain open in this case. ∈ ∗ ∞ Until quite recently, only few existence results were known for p (2N , ). A fruitful approach which has been applied in recent years to treat supercritical problems consists in reducing problem (℘p) to a more general elliptic critical or subcritical problem, either by considering rotational symmetries, or by means of maps which preserve the Laplace operator, or by a combination of both. The aim of this paper is to illustrate this approach by presenting a selection of recent results where it is used to establish existence and multiplicity or to study the concentration behavior of solutions at certain supercritical exponents. To put these results into perspective, we first present some nonexistence results.

2. Nonexistence results ∗ When p =2N a remarkable result obtained by Bahri and Coron [2] establishes the existence of at least one positive solution to problem (℘p)ineverydomain Ω having nontrivial reduced homology with Z/2-coefficients. Moreover, if Ω is invariant under the action of a closed subgroup G of the group O(N) of linear isometries of RN and every G-orbit in Ω is infinite1, the critical problem is known to have infinitely many solutions [6]. Passaseo showed in [23,24] that neither of these conditions is enough to guarantee existence in the supercritical case. He proved the following result. Theorem 2.1. For each 1 ≤ k ≤ N − 3 there is a domain Ω such that (a) Ω has the homotopy type of Sk, (b) Ω is O(k +1)-invariant with infinite O(k +1)-orbits, ≥ ∗ 2(N−k) (c) (℘p) has no solution for p 2N,k := N−k−2 , ∗ (d) (℘p) has infinitely many solutions for p<2N,k. Here O(k + 1) is the group of all linear isometries of Rk+1 acting on the first k + 1 coordinates of a point in RN . Note that 2(N − k) 2∗ := =2∗ N,k N − k − 2 N−k is the critical Sobolev exponent in dimension N −k. It is called the (k+1)-st critical ∗ ∗ ∗ ··· ∗ exponent in dimension N. Note also that 2N < 2N,1 < 2N,2 < < 2N,N−3 =6, ∗ ≥ so 2N,k is supercritical in dimension N if k 1. Passaseo’s domains are defined as Ω:={(y, z) ∈ Rk+1 × RN−k−1 :(|y| ,z) ∈ B},

1Recall that the G-orbit of a point x ∈ Rn is the set Gx := {gx : g ∈ G}. A subset X of Rn is said to be G-invariant if Gx ⊂ X for every x ∈ X, and a function f : X → R is G-invariant if it is constant on every G-orbit of X.

A SUPERCRITICAL ELLIPTIC PROBLEM 3 where B isanyopenball,centeredin(0, ∞) ×{0}, whose closure is contained in the halfspace (0, ∞) × RN−k−1.

N−k−1 R

Passaseo’s result was extended to more general domains by Faya and the authors of this paper in [9]. They also showed that existence may fail even in domains with richer topology. More precisely, they proved the following result. Theorem 2.2. For every ε>0 there is a domain Ω such that (a) Ω has the homotopy type of S1 ×···×S1 (k factors), (b) Ω is O(2) ×···×O(2)-invariant with inﬁnite orbits, ≥ ∗ (c) (℘p) does not have a nontrivial solution for p 2N,k + ε, ∗ (d) (℘p) has inﬁnitely many solutions for p<2N,k. Here S1 stands for the unit circle in R2, and the group O(2) ×···×O(2) with k factors acts on S1 ×···×S1 in the obvious way. Note that there are k cohomology classes in H1(Ω; Z) whose cup-product is nonzero. In fact, the cup-length of Ω is k +1. These domains are of the form (2.1) Ω := {(y1,...,yk,z) ∈ R2 ×···×R2 × RN−2k :(|y1|,...,|yk|,z) ∈ B}, where B is an open ball centered in (0, ∞)k ×{0}, whose closure is contained in (0, ∞)k × RN−2k and whose radius decreases as ε → 0.

N−k−m R

The question whether one can get rid of ε remains open.

Problem 1. Is it true that (℘p) does not have a nontrivial solution in the ≥ ∗ domain Ω deﬁned in (2.1) if p 2N,k and B is an open ball of arbitrary radius? 3. Solutions for higher critical exponents via Hopf maps

A fruitful approach to produce solutions to a supercritical problem (℘p)isto reduce it to some problem of the form (3.1) −div(a(x)∇v)=b(x)|v|p−2v in Θ,v=0 on∂Θ,

4MONICA´ CLAPP AND ANGELA PISTOIA in a bounded smooth domain Θ in Rn, with n := dim Θ < dim Ω = N and the ∈ ∗ ∗ same exponent p.Thus,ifp (2N , 2n], then p is subcritical or critical for problem (3.1) but it is supercritical for (℘p). 3.1. A reduction via Hopf maps. Hopf maps provide a way to obtain such a reduction. For N =2, 4, 8, 16 we write RN = K × K,whereK is either the real numbers R, or the complex numbers C, or the quaternions H, or the Cayley numbers O. N (N/2)+1 The Hopf map hK : R = K × K → R × K = R is given by 2 2 hK(z1,z2)=(|z1| −|z2| , 2z1z2). Topologically, it is just the quotient map of K × K onto its orbit space under the action of SK := {ζ ∈ K : |ζ| =1} given by multiplication on each coordinate, i.e. ζ(y, z):=(ζy,ζz)forζ ∈ SK, (y, z) ∈ K × K. But, regarding our problem, the most relevant property of hK is of geometric nature. It is the fact that hK preserves the Laplace operator. Maps with this property are called harmonic morphisms [3, 30]. The following statement can be derived by straightforward computation or from the general theory of harmonic morphisms.

2 Proposition 3.1. Let Ω ⊂ K be an SK-invariant domain such that 0 ∈/ Ω. Set Θ:=hK(Ω). Then u is an SK-invariant solution to problem (℘p) iﬀ the function v given by u = v ◦ hK is a solution to problem 1 (3.2) −Δv = |v|p−2v in Θ,v=0 on ∂Θ. 2 |x| Note that, if K = C, H or O, then dim Θ = dim K +1 < 2dimK = dim Ω. ∗ ∗ Therefore, p := 2dim K+1 =2N,dim K−1 is critical for (3.2) and supercritical for (℘p). Recently Pacella and Srikanth showed that the real Hopf map provides a one-to- one correspondence between [O(m) × O(m)]-invariant solutions of (℘p)inadomain ΩinR2m and O(m)-invariant solutions of (3.2) in some domain Θ in Rm+1,where O(m)actsonthelastm coordinates of Rm+1 ≡ R × Rm. In [22] they proved the following result. Proposition 3.2. Let N =2m and Ω be an [O(m) × O(m)]-invariant bounded smooth domain in R2m such that 0 ∈/ Ω. Set m Θ:={(t, ζ) ∈ R × R : hR(|y1| , |y2|)=(t, |ζ|) for some (y1,y2) ∈ Ω}.

Then v(t, ζ)=w(t, |ζ|) is an O(m)-invariant solution of problem (3.2) iﬀ u(y1,y2)= w(hR(|y1| , |y2|)) is an [O(m) × O(m)]-invariant solution of problem (℘p). 3.2. Multiplicity results in symmetric domains. The previous proposi- tions suggest to study the critical problem 2∗ −2 ∗ −Δv = b(x) |v| n v in Θ, (℘ ) b v =0 on∂Θ, in a bounded smooth domain Θ in Rn, n ≥ 3, where b : Θ → R is a positive continuous function. This problem is variational but, due to the lack of compactness of the associated energy functional, classical variational methods cannot be applied to establish existence of solutions.

A SUPERCRITICAL ELLIPTIC PROBLEM 5

Under suitable symmetry assumptions compactness is restored: if G is a closed subgroup of the group O(n) of linear isometries of Rn, Θandb are G-invariant, and ∗ every G-orbit in Θ has infinite cardinality, problem (℘b ) is known to have infinitely many G-invariant solutions [6]. This fact, together with Proposition 3.1, provides examples of domains in which problem (℘p) has infinitely many solutions for some higher critical exponents. For example, one has the following result.

Theorem 3.3. Let Θ be a solid of revolution around the z-axis in R3 whose −1 closure does not intersect the z-axis and set Ω:=hC (Θ). Then the supercritical problem

4 (℘ ∗ ) − Δu = |u| u in Ω,u=0 on ∂Ω, 24,1

−1 has inﬁnitely many solutions which are constant on hC {(r cos ϑ, r sin ϑ, t):ϑ ∈ [0, 2π]} for each (r, 0,t) ∈ Θ.

Note that Ω is homeomorphic to Θ × S1. Similar results for problems (℘ ∗ ) 28,3 and (℘ ∗ ) can be derived using the Hopf maps hH and hO. 216,7 ∗ On the other hand, if Θ contains a ﬁnite orbit, problem (℘b )mightnothave a nontrivial solution, as occurs when Θ is a ball centered at the origin in Rn and ≡ ∗ b 1. Conditions which guarantee that problem (℘b ) has a prescribed number of solutions in domains having ﬁnite orbits were recently obtained by Faya and the authors. They proved the following result in [9]. Special cases of it were previusly establishedin[7, 11].

Theorem 3.4. Fix a closed subgroup Γ of O(n) and a bounded smooth Γ- invariant domain D in Rn such that every Γ-orbit in D has inﬁnite cardinality. Assume that b is Γ-invariant. Then there exists a sequence of real numbers (m) with the following property: if Θ ⊃ D and Θ is invariant under the action of a subgroup G of Γ such that

#Gx (3.3) min − >m, ∈ n 2 x Θ b(x) 2

∗ ± then problem (℘b ) has at least m pairs u of G-invariant solutions in Θ; one pair does not change sign and the rest are sign changing.

Here #Gx denotes the cardinality of the G-orbit of x. We illustrate this result with an example. Let D be a torus of revolution around the z-axis in R3, Γ be the group of all rotations around the z-axis, and b(x):= 1 .Fixε>0 smaller than the distance of D to the z-axis. Let G be the 2|x| √ k 2π group generated by the rotation of angle k . If k 2ε>m and Θ is a Gk-invariant domain which contains D whosedistancetothez-axisisatleastε, then every Gk-orbit in Θ has cardinality k and the inequality (3.3) holds true. We call a Gk- invariant domain which contains D and does not intersect the z-axis a k-teething toy.

6MONICA´ CLAPP AND ANGELA PISTOIA

Applying Proposition 3.1 to this example we obtain the following result. Theorem√ 3.5. If Θ is a k-teething toy whose distance to the z-axis is at least ε and k 2ε>m, then the supercritical problem 4 (℘ ∗ ) − Δu = |u| u in Ω,u=0 on ∂Ω, 24,1 −1 has m pairs of SC-invariant solutions in Ω:=hC (Θ); one of them does not change sign and the rest are sign changing. A more general result can be derived from Theorem 3.4, as stated in [9]. 3.3. Existence in domains with thin spherical holes. Next we consider ∗ problem (℘b ) in a punctured domain

Θε := {x ∈ Θ:|x − ξ| >ε}, where Θ is a bounded smooth domain in Rn, n ≥ 3,ξ∈ Θ,ε>0 is small, and b : Θ → R is a positive C2-function. Additionally, we assume that Θ and b are invariant under the action of some closed subgroup G of O(n)andthatξ is a ﬁxed point, i.e. gξ = ξ for all g ∈ G. ≡ 13 ∗ If b 1 Coron showed in [ ] that problem (℘2n ) has a positive solution in Θε for ε small enough. Coron’s proof takes advantage of the fact that the variational functional associated to problem (℘ ∗ ) satisﬁes the Palais-Smale condition between 2n the ground state level and twice that level. This is not true anymore when b ≡ 1, so Coron’s argument does not carry over to the nonautonomous case. A method which has proved to be very successful in dealing with critical problems which involve small perturbations of the domain is the Lyapunov-Schmidt reduction method, see e.g. [17] and the references therein. This method was used by Faya and the authors in [10] to prove the following result. Theorem 3.6. If ∇b(ξ) =0 then, for ε small enough, ∗ ∗ 2 −2 − | | n (℘b ) Δv = b(x) v v in Θε,v=0 on ∂Θε, has a positive G-invariant solution vε in Θε which concentrates at the boundary of the hole and blows up at ξ as ε → 0. Note that G may be the trivial group, so this result is true in a non-symmetric setting and, combined with Proposition 3.1, yields solutions to supercritical problems concentrating around a spherical hole, see [10]. But we may also combine it with Proposition 3.2 as follows: Let N =2m, m ≥ 2, Ωbean[O(m) × O(m)]-invariant bounded smooth domain in RN ≡ Rm × Rm such that 0 ∈/ Ω, and ξ ∈ Ω ∩ (Rm ×{0}) . For ε>0 small enough set

Ωε := {x ∈ Ω:dist(x, Sξ) >ε},

A SUPERCRITICAL ELLIPTIC PROBLEM 7

m where Sξ := {(x, 0) ∈ R ×{0} : |x| = |ξ|}. The following result, obtained by combining Theorem 3.6 with Proposition 3.2, was established in [10]. Theorem 3.7. For each ε small enough the supercritical problem 4/(m−1) (℘ ∗ ) − Δu = |u| u in Ω ,v=0 on ∂Ω , 22m,m−1 ε ε has a positive [O(m) × O(m)]-invariant solution uε which concentrates along the set {x ∈ Ω:dist(x, Sξ)=ε} and blows up at the (m − 1)-dimensional sphere Sξ as ε → 0.

4. Solutions for higher critical exponents via rotations Supercritical problems in domains obtained through rotations can be reduced to subcritical or critical problems as follows.

4.1. A reduction via rotations. Fix k1,...,km ∈ N and set k := k1 + ···+ km. If N ≥ k + m let 1 m k +1 k +1 N−k−m 1 m Ω:={(y ,...,y ,z) ∈ R 1 ×···×R m × R : y ,...,|y | ,z ∈ Θ}, where Θ is a bounded smooth domain in RN−k whose closure is contained in (0, ∞)m × RN−k−m.

N−k−m R

Each point ξ ∈ Θ gives rise to a subset 1 m i (4.1) Tξ := {(y ,...,y ,z) ∈ Ω: y = ξi,z=(ξm+1,...,ξN−k)} of Ω which is homeomorphic to the product of spheres Sk1 ×···×Skm . A straightforward computation yields the following result. Proposition 4.1. A function u of the form u(y1,...,ym,z)=v(y1 ,...,|ym| ,z) is a solution of problem (℘p) iﬀ v is a solution of (4.2) −div(a(x)∇v)=a(x)|v|p−2v in Θ,v=0on ∂Θ,

k1 ··· km with a(x1,...,xN−k):=x1 xm . 4.2. Multiplicity results in domains obtained by rotation. Wei and Yan considered domains as above, with m = 1, where Θ is invariant under the action of the group O(2) × O(1)N−k−3 on the last N − k − 1 coordinates of RN−k, i.e.

(s, r cos θ, r sin θ, x3,...,xN−k) ∈ Θ for all θ ∈ (0, 2π)if(s, r, 0,...,xN−k) ∈ Θ, (x1,,...,−xi,,...,xN−k)∈Θif(x1,...,xi,...,xN−k)∈Θandi=4,...,N − k. They proved the following result in [29].

8MONICA´ CLAPP AND ANGELA PISTOIA Theorem 4.2. Let N ≥ 5. Assume that Θ is O(2) × O(1)N−k−3 -invariant and that there is a point (s∗,r∗) ∈S:= {(s, r) ∈ R2 :(s, r, 0,...,0) ∈ ∂Θ}, which is a strict local minimum (or a strict local maximum) of the distance of S ∗ ∗ ∗ to {0}×R. Set ξj := (s ,r cos (2πj/) ,r sin (2πj/) , 0,...,0). Then, for large enough ∈ N, problem (℘ ∗ ) has a solution u with positive layers; which 2N,k ⊂ − concentrate along each of the k-dimensional spheres Tξj ∂Ω,j=0,..., 1, and blow up at ∂Ω as →∞. They derived this result from Proposition 4.1 after proving the existence of -multibubble solutions for the critical problem ∗ ∗ − − ∇ | |2N−k 2 (℘a,a) div(a(x) v)=a(x) v v in Θ,v=0 on ∂Θ, which concentrate at the points ξj ∈ ∂Θ. A precise description of the solutions is given in [29]. In [20] Kim and Pistoia considered domains with thin k-dimensional holes. More precisely, for some ﬁxed ξ ∈ Θ, they considered

Ωε := {x ∈ Ω:dist(x, Tξ) >ε}, with Tξ as in (4.1) and ε>0 suﬃciently small. Note that Ωε is obtained by rotating the punctured domain Θε := {x ∈ Θ:|x − ξ| >ε} as described in subsection 4.1. For N − k ≥ 4 they proved the existence of towers of bubbles with alternating signs ∗ around ξ for the anisotropic problem (℘a,a)inΘε, thus extending a previous result by Ge, Musso and Pistoia [17] for the autonomous problem a ≡ 1. The number of bubbles increases as ε → 0. Combining this result with Proposition 4.1 they obtained the following one.

Theorem 4.3. Let N ≥ k +4. Then, for every ∈ N there exists ε > 0 such that, for each ε ∈ (0,ε), the supercritical problem ∗ − 2N,k 2 −Δu = |u| u in Ωε,v=0 on ∂Ωε, has a solution uε with layers of alternating signs, which concentrate with diﬀerent rates along the boundary of the tubular neighborhood of radius ε of Tξ and blow up at Tξ as ε → 0. A precise description of the solutions can be found in [20]. The existence of a prescribed number of solutions to problem (℘ ∗ )indo- 2N,k mains obtained via rotations was recently established in [8] under some symmetry assumptions.

5. Concentration along manifolds at the higher critical exponents 5.1. Approaching the higher critical exponents from below. For do- ∈ ∗ mains as those described in subsection 4.1 and p (2, 2N,k)problem( ℘p) has inﬁn- itely many nontrivial solutions of the form u(y1,...,ym,z)=v(y1 ,...,|ym| ,z). This follows immediately from Proposition 4.1 using standard variational methods because problem (4.2) is subcritical and the domain Θ is bounded. Existence and nonexistence results in some unbounded domains are also available [12]. So the question is whether one can establish existence of solutions to (℘p)which → ∗ exhibit a certain concentration behavior as p 2N,k.

A SUPERCRITICAL ELLIPTIC PROBLEM 9

For the slightly subcritical problem (℘ ∗ − ) positive and sign changing solu- 2N ε tions uε which blow up at one or several points in Ω as ε → 0 have been obtained e.g. in [4, 26, 28]. Recently, del Pino, Musso and Pacard [16] considered the case in which p approaches the second critical exponent from below. They showed that, if N ≥ 8and∂Ω contains a nondegenerate closed geodesic Γ with negative inner normal curvature then, for every ε>0 small enough, away from an explicit discrete set of values, problem (℘ ∗ − ) has a positive solution u which concentrates and 2N,1 ε ε blowsupatΓasε → 0. It is natural to ask whether similar concentration phenomena can be observed ∗ as p aproaches the (k + 1)-st critical exponent 2N,k from below, i.e. whether there are domains in which problem (℘ ∗ − ) has a solution u which concentrates and 2N,k ε ε blows up at a k-dimensional submanifold of Ω as ε → 0. Ackermann, Kim and the authors have given positive answers to this question for domains Ω as in subsection 4.1. Let K be the set of all nondegenerate critical points ξ of the restriction of k1 ··· km ∇ the function a(x1,...,xN−k):=x1 xm to ∂Θ such that a(ξ) points into the interior of Θ, and let Tξ be the set deﬁned in (4.1). The following result was proved in [1].

Theorem 5.1. For any subset {ξ1,...,ξ} of K and 1 ≤ m ≤ there exists ε > 0 such that, for each ε ∈ (0,ε ), problem (℘ ∗ − ) has a solution u with m 0 0 2N,k ε ε positive layers and − m negative layers; which concentrate at the same rate and → blow up along one of the sets Tξi as ε 0. Sign changing solutions are also available. Statement (a) in the following the- oremwasprovedin[1] and statement (b) was proved in [19].

N−k Theorem 5.2. Assume there exist ξ0 ∈Kand τ1,...,τN−k−1 ∈ R such that the set {∇a(ξ0),τ1,...,τN−k−1} is orthogonal and Θ and a are invariant with respect to the reﬂection i on the hyperplane through ξ0 which is orthogonal to τi, for each i =1,...,N − k − 1. Then the following statements hold true: (a) For each ε>0 small enough problem (℘ ∗ − ) has a sign changing solu- 2N,k ε tion uε with one positive and one negative layer, which concentrate at the → same rate along Tξ0 andblowupatTξ0 as ε 0. (b) If k ≤ N − 4 then, for any integer ≥ 2, there exists ε > 0 such that, for ε ∈ (0,ε ), problem (℘ ∗ − ) has a solution u with layers of alternating 2N,k ε ε

signs which concentrate at diﬀerent rates along Tξ0 andblowupatTξ0 as ε → 0. These results follow from Proposition 4.1 once the corresponding statements for the slightly subcritical problems 2∗ −ε−2 −div(a(x)∇v)=a(x)|v| N−k v in Θ,v=0 on ∂Θ, have been established. This is done in [1, 19], where a precise description of the solutions is given. The following questions were raised in [1]: Problem 2. Can statement (a) in Theorem 5.2 be improved to establish existence of solutions with layers of alternating signs which concentrate at the same → ≥ rate along Tξ0 and blow up at Tξ0 as ε 0, for any 2? Problem 3. Does Theorem 5.2 hold true without the symmetry assumption?

10 MONICA´ CLAPP AND ANGELA PISTOIA

Recently, Pacella and Pistoia considered the case in which N =2m and Ω is the annulus Ω := {x ∈ RN :0

5.2. Approaching the higher critical exponents from above. For the slightly supercritical problem (℘ ∗ ),ε>0, in a domain Ω with nontrivial topol- 2N +ε ogy del Pino, Felmer and Musso established the existence of a positive solution uε with two bubbles which concentrate at two diﬀerent points ξ1,ξ2 ∈ Ωasε → 0[14]. Solutions with more that two bubbles are also available, see e.g. [4, 15, 25]. For ε suﬃciently small solutions with only one bubble do not exist [5]. The following problem is fully open to investigation.

Problem 4. Are there domains Ω in which (℘ ∗ ) has positive or sign 2N,k+ε changing solutions uε whichconcentrateandblowupatk-dimensional manifolds as ε → 0?

References [1] Nils Ackermann, Mónica Clapp, and Angela Pistoia, Boundary clustered layers near the higher critical exponents, J. Differential Equations 254 (2013), no. 10, 4168–4193, DOI 10.1016/j.jde.2013.02.015. MR3032301 [2] A. Bahri and J.-M. Coron, On a nonlinear elliptic equation involving the critical Sobolev exponent: the effect of the topology of the domain, Comm. Pure Appl. Math. 41 (1988), no. 3, 253–294, DOI 10.1002/cpa.3160410302. MR929280 (89c:35053) [3] Paul Baird and John C. Wood, Harmonic morphisms between Riemannian manifolds,Lon- don Mathematical Society Monographs. New Series, vol. 29, The Clarendon Press, Oxford University Press, Oxford, 2003. MR2044031 (2005b:53101) [4] Thomas Bartsch, Anna Maria Micheletti, and Angela Pistoia, On the existence and the profile of nodal solutions of elliptic equations involving critical growth, Calc. Var. Partial Dif- ferential Equations 26 (2006), no. 3, 265–282, DOI 10.1007/s00526-006-0004-6. MR2232205 (2007b:35102) [5]M.BenAyed,K.ElMehdi,O.Rey,andM.Grossi,A nonexistence result of single peaked solutions to a supercritical nonlinear problem, Commun. Contemp. Math. 5 (2003), no. 2, 179–195, DOI 10.1142/S0219199703000951. MR1966257 (2004k:35140) [6] Mónica Clapp, A global compactness result for elliptic problems with critical nonlinearity on symmetric domains, Nonlinear equations: methods, models and applications (Bergamo, 2001), Progr. Nonlinear Differential Equations Appl., vol. 54, Birkhäuser, Basel, 2003, pp. 117–126. MR2023237 (2004j:35093) [7] Mónica Clapp and Jorge Faya, Multiple solutions to the Bahri-Coron problem in some domains with nontrivial topology, Proc. Amer. Math. Soc. 141 (2013), no. 12, 4339–4344, DOI 10.1090/S0002-9939-2013-12043-5. MR3105875 [8] M. Clapp, J. Faya, Multiple solutions to anisotropic critical and supercritical problems in symmetric domains. Progr. Nonlinear Differential Equations Appl 86 (2015), 99-120, DOI 10.1007/978-3-319-19902-3 8. [9] Mónica Clapp, Jorge Faya, and Angela Pistoia, Nonexistence and multiplicity of solutions to elliptic problems with supercritical exponents, Calc. Var. Partial Differential Equations 48 (2013), no. 3-4, 611–623, DOI 10.1007/s00526-012-0564-6. MR3116025 [10] M. Clapp, J. Faya, and A. Pistoia, Positive solutions to a supercritical elliptic problem which concentrate along a thin spherical hole. J. Anal. Math., 126 (2015), 341-357, DOI: 10.1007/s11854-015-0020-6. MR3358036

A SUPERCRITICAL ELLIPTIC PROBLEM 11

[11] Mónica Clapp and Filomena Pacella, Multiple solutions to the pure critical exponent problem in domains with a hole of arbitrary size,Math.Z.259 (2008), no. 3, 575–589, DOI 10.1007/s00209-007-0238-9. MR2395127 (2009f:35076) [12] M. Clapp, A. Szulkin, A supercritical elliptic problem in a cylindrical shell. Progr. Nonlinear Differential Equations Appl 85 (2014), 233–242. [13] Jean-Michel Coron, Topologie et cas limite des injections de Sobolev (French, with English summary),C.R.Acad.Sci.ParisSér. I Math. 299 (1984), no. 7, 209–212. MR762722 (86b:35059) [14] Manuel del Pino, Patricio Felmer, and Monica Musso, Two-bubble solutions in the supercritical Bahri-Coron’s problem, Calc. Var. Partial Differential Equations 16 (2003), no. 2, 113–145, DOI 10.1007/s005260100142. MR1956850 (2004a:35079) [15] Manuel del Pino, Patricio Felmer, and Monica Musso, Multi-bubble solutions for slightly super-critical elliptic problems in domains with symmetries, Bull. London Math. Soc. 35 (2003), no. 4, 513–521, DOI 10.1112/S0024609303001942. MR1979006 (2004c:35136) [16] Manuel del Pino, Monica Musso, and Frank Pacard, Bubbling along boundary geodesics near the second critical exponent, J. Eur. Math. Soc. (JEMS) 12 (2010), no. 6, 1553–1605, DOI 10.4171/JEMS/241. MR2734352 (2012a:35115) [17] Yuxin Ge, Monica Musso, and Angela Pistoia, Sign changing tower of bubbles for an elliptic problem at the critical exponent in pierced non-symmetric domains,Comm.PartialDifferen- tial Equations 35 (2010), no. 8, 1419–1457, DOI 10.1080/03605302.2010.490286. MR2754050 (2011k:35070) [18] Jerry L. Kazdan and F. W. Warner, Remarks on some quasilinear elliptic equations, Comm. Pure Appl. Math. 28 (1975), no. 5, 567–597. MR0477445 (57 #16972) [19] Seunghyeok Kim and Angela Pistoia, Boundary towers of layers for some supercritical problems, J. Differential Equations 255 (2013), no. 8, 2302–2339, DOI 10.1016/j.jde.2013.06.017. MR3082463 [20] Seunghyeok Kim and Angela Pistoia, Supercritical problems in domains with thin toroidal holes, Discrete Contin. Dyn. Syst. 34 (2014), no. 11, 4671–4688, DOI 10.3934/dcds.2014.34.4671. MR3223824 [21] F. Pacella, A. Pistoia, Bubble concentration on spheres for supercritical elliptic problems. Progr. Nonlinear Differential Equations Appl 85 (2014), 323-340. [22] Filomena Pacella and P. N. Srikanth, A reduction method for semilinear elliptic equations and solutions concentrating on spheres,J.Funct.Anal.266 (2014), no. 11, 6456–6472, DOI 10.1016/j.jfa.2014.03.004. MR3192458 [23] Donato Passaseo, Nonexistence results for elliptic problems with supercritical nonlinearity in nontrivial domains, J. Funct. Anal. 114 (1993), no. 1, 97–105, DOI 10.1006/jfan.1993.1064. MR1220984 (94m:35118) [24] Donato Passaseo, New nonexistence results for elliptic equations with supercritical nonlinearity, Differential Integral Equations 8 (1995), no. 3, 577–586. MR1306576 (95j:35086) [25] Angela Pistoia and Olivier Rey, Multiplicity of solutions to the supercritical Bahri-Coron’s problem in pierced domains, Adv. Differential Equations 11 (2006), no. 6, 647–666. MR2238023 (2007f:35102) [26] Angela Pistoia and Tobias Weth, Sign changing bubble tower solutions in a slightly subcritical semilinear Dirichlet problem (English, with English and French summaries), Ann. Inst. H. Poincaré Anal. Non Linéaire 24 (2007), no. 2, 325–340, DOI 10.1016/j.anihpc.2006.03.002. MR2310698 (2008c:35082) [27] S.I. Pohozhaev, Eigenfunctions of the equation Δu+λf(u)=0. Soviet Math. Dokl. 6 (1965), 1408-1411. [28] Olivier Rey, Blow-up points of solutions to elliptic equations with limiting nonlinearity,Dif- ferential Integral Equations 4 (1991), no. 6, 1155–1167. MR1133750 (92i:35056) [29] Juncheng Wei and Shusen Yan, Infinitely many positive solutions for an elliptic problem with critical or supercritical growth (English, with English and French summaries), J. Math. Pures Appl. (9) 96 (2011), no. 4, 307–333, DOI 10.1016/j.matpur.2011.01.006. MR2832637 [30] John C. Wood, Harmonic morphisms between Riemannian manifolds, Modern trends in geometry and topology, Cluj Univ. Press, Cluj-Napoca, 2006, pp. 397–414. MR2250232 (2007h:53096)

12 MONICA´ CLAPP AND ANGELA PISTOIA

Instituto de Matematicas,´ Universidad Nacional Autonoma´ de Mexico,´ Circuito Exterior, C.U., 04510 Mexico´ D.F., Mexico E-mail address: [email protected] Dipartimento di Metodi e Modelli Matematici, Universita´ di Roma “La Sapienza”, via Antonio Scarpa 16, 00161 Roma, Italy E-mail address: [email protected]

Contemporary Mathematics Volume 656, 2016 http://dx.doi.org/10.1090/conm/656/13103

Min-max theory of minimal surfaces and applications

Fernando C. Marques and Andr´e Neves

Abstract. This paper gives a brief account of some recent results of the authors: the proof of the Willmore conjecture for surfaces; the proof of the Freedman-He-Wang conjecture for links (jointly with Agol); the proof of the existence of inﬁnitely many minimal hypersurfaces in manifolds of positive Ricci curvature.

1. Introduction This paper gives a brief account of some recent results of the authors: the proof of the Willmore conjecture for surfaces; the proof of the Freedman-He-Wang conjecture for links (jointly with Agol); the proof of the existence of infinitely many minimal hypersurfaces in manifolds of positive Ricci curvature. These theorems are aplications of the min-max theory for the area functional, more specifically the theory developed in the early 1980s by Almgren and Pitts. Minimal surfaces have been studied in Differential Geometry since the pioneering work of Lagrange (1762). They are defined as surfaces that locally minimize the area, a variational principle satisfied precisely by surfaces with zero mean curvature. The problem of finding a closed embedded minimal surface in a compact Riemannian three-manifold is difficult in general because these surfaces are not necessarily area-minimizing. The idea of producing minimal surfaces by min-max methods has its roots in the work of Birkhoff [5], who studied the problem of existence of closed geodesics in Riemannian 2-spheres, a foundational question raised previously by Poincaré[22]. Birkhoff’s ideas later inspired the celebrated critical point theories of Morse and of Lusternik and Schnirelmann. The first steps to attack the problem of existence of higher-dimensional minimal submanifolds were taken by Almgren ([2], [3]). He developed a general scheme to produce minimal varieties in Riemannian manifolds. The question of regularity of these objects was left open, being solved later by his Ph.D. student Jon Pitts [23] in the important case of codimension one and ambient dimension less than or equal to 6. Their combined works form what we refer to as the Almgren-Pitts min-max theory. The regularity theory was then extended by Schoen and Simon [26]. Putting everything together the main application was up until very recently:

The first author was partly supported by CNPq-Brazil, FAPERJ and UniversitéParis-Est (Labex Bézout). The second author was partly supported by Marie Curie IRG Grant and ERC Start Grant.

c 2016 American Mathematical Society 13

14 FERNANDO C. MARQUES AND ANDRE´ NEVES

Theorem 1.1. Let (M n,g) be an n-dimensional compact Riemannian manifold, with 3 ≤ n ≤ 7. Then there exists a smooth, embedded, closed minimal hypersurface Σn−1 ⊂ M. If n ≥ 8, then such Σ can be constructed with a possible singular set of codimension 7. The minimal hypersurface Σ can be obtained by applying the min-max technique to the class of sweepouts of M, topologically nontrivial one-parameter families of hypersurfaces. These are the families of surfaces {Σt}t∈[0,1] such that we can write Σt = ∂Ωt with Ωt varying continuously (in volume sense), where Ω0 =0 and Ω1 = M. The family of level sets {x ∈ M : f(x)=t} of a Morse function f : M → [0, 1] is an example of a sweepout. Let us informally describe this min-max theory. We restrict ourselves to the 3 case of three-dimensional ambient spaces for simplicity. We denote by Z2(M )the space of integral 2-cycles in M (integral 2-currents with zero boundary). We would n 3 like to run min-max with maps Φ : I → Z2(M ) continuous in the flat topology, and defined on the n-dimensional cube In =[0, 1]n. If Π is the homotopy class of Φ relative to the boundary ∂In,wedefinethewidthofΠby L(Π) = inf{L(Φ):Φ ∈ Π}, n where L(Φ )=sup{area(Φ (x)) : x ∈ I }. Any sequence {φi}i ⊂ Π with L(φi) → L(Π) is called an optimal sequence. The prototypical min-max theorem is:

Min-max Theorem: If L(Π) > sup{area(Φ(x)) : x ∈ ∂In}, then there exists a smooth embedded minimal surface Σ (possibly disconnected, with integer multiplicities) in M so that L(Π) = area(Σ). Moreover, if {φi}i ⊂ Π is an optimal sequence n then we can choose Σ so that for a subsequence {j}⊂{i} and some xj ∈ I we have that Σ is the limit of φj (xj) in varifold sense. Given an orientable compact Riemannian three-manifold M, we can start from a Morse function f : M → [0, 1] and deﬁne Φ(t)={x ∈ M : f(x)=t}, t ∈ [0, 1]. We denote by Π1 the homotopy class (relative to {0, 1}) of Φ. Since for any Φ ∈ Π1, Φ (t)=∂Ω (t), there exists t0 ∈ [0, 1] such that vol(Ω (t0)) = vol(M)/2. Therefore the isoperimetric inequality tells us that L(Π1) > 0. By the Min-max Theorem there will be a closed embedded minimal surface Σ ⊂ M such that L(Π1) = area(Σ). This is the hypersurface Σ of Theorem 1.1. If M is the unit three-sphere we can take a standard foliation by round spheres: 3 Φ(t)={x ∈ S : x4 =1−2t}, t ∈ [0, 1], to conclude that L(Π1) ≤ 4π.Butthearea of any closed minimal surface in S3 is at least 4π, with equality only if the surface is a great sphere. Therefore the Almgren-Pitts min-max minimal surface of S3 is the equator with area L(Π1)=4π. The next theorem follows from the previous discussion:

3 3 1.1. 4π Theorem. Let Φ:I →Z2(S ) be a sweepout of S . Then there exists y ∈ [0, 1] such that area(Φ(y)) ≥ 4π.

MIN-MAX THEORY OF MINIMAL SURFACES AND APPLICATIONS 15

There are infinitely many closed minimal surfaces in S3 (Lawson [18]), and among those, the simplest after the equator is the Clifford torus 1 1 Σ=ˆ S1(√ ) × S1(√ ). 2 2 This minimal surface has area 2π2, index 5, and, in fact, was characterized by its index in a theorem of Urbano: Theorem 1.2. ([30])LetΣ ⊂ S3 be a smooth, closed minimal surface with index(Σ) ≤ 5.ThenΣ is either a great sphere or the Clifford torus, up to ambient isometries.

The question we posed ourselves, and that it turned out to be key to the solution of the Willmore conjecture, was whether it is possible to produce the Clifford torus by min-max methods. We have answered this question affirmatively by working with a certain class of five-parameter sweepouts. We consider maps 5 3 5 Φ:I →Z2(S ) defined on the 5-cube I that satisfy: (1) Φ(x, 0) = Φ(x, 1) = 0 (trivial surface) for any x ∈ I4, 3 (2) {Φ(x, t)}t∈[0,1] is the standard sweepout of S by oriented round spheres centered at Q(x) ∈ S3, for any x ∈ ∂I4, 4 (3) Φ(x, 1/2) = ∂Bπ/2(Q(x)), for any x ∈ ∂I . If deg(Q) = 0, this map Φ has the crucial property that its restriction to ∂I4 ×{1/2} is a homotopically nontrivial map into the space of oriented great spheres. We proved in [20]:

2 5 3 1.2. 2π Theorem. ([20]) Let Φ:I →Z2(S ) be a continuous map in the ﬂat topology satisfying the properties (1)-(3) above, with center map Q : ∂I4 → S3. If deg(Q) =0 , then there must exist y ∈ I5 with area(Φ(y)) ≥ 2π2.

We use the 2π2 Theorem to solve both the Willmore conjecture (Section 2) and the Freedman-He-Wang conjecture (Section 3). In Section 4, we give a brief description of our proof of the existence of infinitely many minimal hypersurfaces in manifolds of positive Ricci curvature. This uses min-max theory applied to families previously studied by Gromov and Guth. We note that Yau conjectured in [32](first problem in the Minimal Surfaces section) that every compact Riemannian three- manifold admits an infinite number of smooth, closed, immersed minimal surfaces. Understanding the behavior of the area, index, multiplicity and distribution of the minimal hypersurfaces we construct are important questions.

2. Willmore conjecture Given a closed surface Σ ⊂ R3 of some genus g, its Willmore energy is deﬁned by W(Σ) = H2dΣ, Σ where H =(k1 + k2)/2 denotes the mean curvature (k1,k2 are the principal curva- tures of Σ).

16 FERNANDO C. MARQUES AND ANDRE´ NEVES

This functional is remarkably symmetric. It follows immediately from the definition that it is invariant under ambient isometries and scalings, but it is actually also invariant under inversions. Indeed, using the Gauss-Bonnet Theorem, (k − k )2 1 H2dΣ= K + 1 2 dΣ=2πχ(Σ) + |A˚|2dΣ, Σ Σ 4 2 Σ where K = k1k2 denotes the Gauss curvature and A˚ is the trace-free second fundamental form. The fact that the expression |A˚|2dΣ is invariant under inversions was known back in the 1920s by Blaschke [6]andThomsen[29]. Hence W (F (Σ)) = W (Σ) for any conformal transformation F of three-space. The idea that the round shape should provide the best immersion of a two- sphere in R3, and that the Willmore energy should measure how good an immersion is, are compatible with the following theorem proven by Willmore in the early 1960s:

2.1. Theorem. (Willmore) Let Σ beasmoothclosedsurfaceinR3.Then W(Σ) ≥ 4π, and equality holds if and only if Σ is a round sphere. Motivated by this result, and after analyzing the particular case of tori of revolution with circular section, Willmore made the following conjecture:

Willmore Conjecture. ([31], 1965): If Σ ⊂ R3 is a torus, then W(Σ) ≥ 2π2. √ √ If Σ 2 denotes the torus obtained by rotation of a circle with center at distance 2 of the axis of revolution and radius 1, then one can compute W √ 2 (Σ 2)=2π . Because of the conformal invariance of the energy, and given that the stereographic projection is conformal, it is extremely convenient to formulate the conjecture in terms of surfaces in the three-sphere S3 instead. If π : S3 \{p}→R3, p ∈ S3, denotes a stereographic projection, and Σ ⊂ S3 \{p}, we can calculate the energy of its projection Σ=˜ π(Σ) ⊂ R3: H˜ 2dΣ=˜ (1 + H2) dΣ, Σ˜ Σ where H now denotes the mean curvature of Σ with respect to the spherical geometry. Hence we define the Willmore energy of Σ ⊂ S3 by the formula: W(Σ) = (1 + H2) dΣ. Σ The relation with minimal surfaces becomes apparent: first, W(Σ) ≥ area(Σ), and W(Σ) = area(Σ) if and only if Σ is a minimal surface. Secondly, the torus√ of √ ˆ 1 × Willmore√ Σ 2 is a stereographic projection of the Clifford torus Σ=S (1/ 2) 1 ⊂ 3 2 ⊂ 3 S (1/ 2) S , the simplest minimal surface after the equator S1 (0) S . Notice that the area of the equator is 4π, while the area of the Clifford torus is 2π2.In [18], Lawson gave infinitely many examples of closed minimal surfaces in S3. There is a long list of results associated to the Willmore conjecture (see [20]for references). In particular, Li and Yau [19] proved that any surface that contains at least one self-intersection should have energy bounded below by 8π. This means that we are allowed to assume that Σ is embedded. The Willmore conjecture follows from our main result in [20]:

MIN-MAX THEORY OF MINIMAL SURFACES AND APPLICATIONS 17

Theorem 2.1. Let Σ ⊂ S3 be a closed, embedded smooth surface with genus g ≥ 1.ThenW(Σ) ≥ 2π2,andW(Σ) = 2π2 if and only if Σ is a conformal image of the Clifford torus. Remark: It is known that minimizers of the Willmore energy among surfaces of a given genus g always exist. This was first proven for g =1byL.Simon[27], then extended to higher genus by Bauer and Kuwert [4](seealso[16]). The minimum 3 Willmore energy βg among all orientable closed surfaces of genus g in R is less 2 than 8π, and converges to 8π as g →∞[17]. Our result implies that β1 =2π and 2 βg > 2π for every g ≥ 2. The proof of Theorem 2.1 is based on the construction, for each embedded 3 5 3 closed surface Σ of genus g ≥ 1inS , of a suitable family Φ : I →Z2(S )that satisfies the assumptions of the 2π2 Theorem and is such that • the center map Q satisfies deg(Q) = g, • and area(Φ(x)) ≤W(Σ) for each x ∈ I5. Let us now sketch the construction of the canonical family. Let B4 be the unit ball. For every v ∈ B4 we consider the conformal map (1 −|v|2) F : S3 → S3,F(x)= (x − v) − v. v v |x − v|2 3 If v =0then Fv is a centered dilation of S that fixes v/|v| and −v/|v|. Let S3 \ Σ=A ∪ A∗, A and A∗ the connected components, and let N be the unit normal vector to Σ pointing into A∗. We consider the following images under Fv: ∗ ∗ Av = Fv(A),Av = Fv(A )andΣv = Fv(Σ) = ∂Av. The unit normal vector to Σv is given by Nv = DFv(N)/|DFv(N)|. We associate, to each smooth embedded closed surface Σ ⊂ S3, a canonical five-dimensional family of surfaces: 3 4 Σ(v,t) = ∂ x ∈ S : dv(x)

Proposition 2.2. Let Σt, t ∈ (−π, π), be an equidistant surface of an embedded closed surface Σ ⊂ S3.Then

area(Σt) ≤W(Σ). Moreover, if Σ is not a geodesic sphere and

area(Σt)=W(Σ), then t =0and Σ is a minimal surface. By conformal invariance of the Willmore energy, we get that area Σ(v,t) ≤W(Σ) for every (v, t) ∈ B4 × (−π, π). Moreover, if Σ is not a geodesic sphere and area Σ(v,t) = W(Σ), then t =0andΣv is a minimal surface.

18 FERNANDO C. MARQUES AND ANDRE´ NEVES

3 Next we must analyze the behavior of the surfaces Σ(v,t) as v → S .Thisis important because to apply min-max theory we need to have a continuous family of surfaces defined up to the boundary of the parameter space. We denote by XΔY =(X \ Y ) ∪ (Y \ X) the symmetric difference of X and Y . → ∈ The subtle case is to understand what happens to Fvn (Σ) when vn v = p Σ. We need to introduce some notation. Let 2 { ∈ R2 | | ≥ } D+(r)= s =(s1,s2) : s 0 sufficiently small so that Λ : Σ D+(3ε) B given by

Λ(p, s)=(1− s1)(cos(s2)p + sin(s2)N(p)) 4 is a diﬀeomorphism onto a tubular neighbourhood of Σ in B .WesetΩr =Λ(Σ× 2 D+(r)). If vn → v = p ∈ Σ, we can write vn =Λ(pn, (sn1,sn2)), with pn → p and |sn|→

0. The idea is to show that the limit of Avn = Fvn (A) is the same limit of Fvn (Bpn ), where B = B√4 (−N(p )) ∩ S3. After explicitly computing the conformal images pn 2 n of balls in S3 one arrives at the statement below, which in particular implies that → 4 Avn must subsequentially converge to a geodesic ball as vn ∂B : For p ∈ S3 and k ∈ [−∞, +∞], we set Q = − √ k p − √ 1 N(p) ∈ S3 and p,k 1+k2 1+k2 π − ∈ rk = 2 arctan k [0,π]. 4 Proposition 2.3. Consider a sequence (vn,tn) ∈ B × (−π, π) converging to 4 (v, t) ∈ B × [−π, π]. (i) If v ∈ B4 then lim vol A(v ,t )ΔA(v,t) =0. n→∞ n n (ii) If v ∈ A then lim vol A(v ,t )ΔBπ+t(v) =0 n→∞ n n (iii) If v ∈ A∗ then lim vol A(v ,t )ΔBt(−v) =0 n→∞ n n (iv) If v = p ∈ Σ and s v =Λ(p , (s ,s )) with lim n2 = k ∈ [−∞, ∞], n n n1 n2 →∞ n sn1 then

lim vol A(v ,t )ΔBrk+t(Qp,k) =0. n→∞ n n Based on the previous result, we can reparametrize the canonical family in such 4 a way that it can be continuously extended to B × [−π, π]. Let φ :[0, 3ε] → [0, 1] be a smooth function such that • φ([0,ε]) = 0, • φ is strictly increasing in [ε, 2ε], • φ([2ε, 3ε]) = 1. 4 4 We deﬁne T : B → B by ∈ 4 \ T (v)= v if v B Ω3ε Λ(p, φ(|s|)s)ifv =Λ(p, s) ∈ Ω3ε.

MIN-MAX THEORY OF MINIMAL SURFACES AND APPLICATIONS 19

Notice that the map T is continuous and collapses Ωε onto Σ by preserving the 4 4 ratio sn2/sn1.Moreover,T : B \ Ω → B is a homeomorphism. 4 For each v ∈ B \ Ωε, we put

C(v, t)=Σ(T (v),t). 4 The last proposition then implies that as v → ∂(B \ Ωε), the surface C(v, t) converges to some geodesic sphere (round sphere). We extend C to Ωε by making it constant along the curves s1 → Λ(p, s1,s2). By construction, we have that 4 C(v, t) is continuous (in volume sense) in B × [−π, π] and that C(v, t)=∂Br(v)+t(Q(v)) for v ∈ ∂B4,wherethecenter map Q : S3 → S3 is given by ⎧ ∗ ⎨ −T (v)ifv ∈ A \ Ω T (v)ifv ∈ A \ Ω Q(v)=⎩ √ − s − ε2−s2 ∈ − ε p ε N(p)ifv =cos(s)p + sin(s)N(p),s [ ε, ε], and r : S3 → [0,π] is the function given by ⎧ ∗ ⎨⎪ 0ifv ∈ A \ Ωε, π if v ∈ A \ Ω , r(v)=⎪ ε ⎩ π − arctan(√ s2 )ifv =Λ(p, s) ∈ S3 ∩ Ω . 2 2− 2 ε ε s2 The main topological ingredient is the discovery that the information of the genus of the original surface Σ can be extracted from the topological properties of the canonical family at the boundary. We proved in [20]that: Theorem 2.4. deg(Q) = genus(Σ). 4 Since B × [−π, π] is homeomorphic to I5,wecanreparametrizeoncemoreto 5 3 4 3 get a family Φ : I →Z2(S ) with center map Q : ∂I → S satisfying deg(Q)=g. Then area(Φ(y)) ≥ 2π2 for some y ∈ I5 by the 2π2 Theorem, hence W(Σ) ≥ area(Φ(y)) ≥ 2π2. The rigidity case can also be proven by analyzing the equality case in Proposition 2.2.

3. Links in three-space We describe now a solution to the problem of determining the best 2-component 3 link in three-space. A 2-component link in R is a pair (γ1,γ2) of rectifiable curves 1 3 1 1 γi : S → R , i =1, 2, such that γ1(S ) ∩ γ2(S )=∅.TheMöbius cross energy of the link (γ ,γ )isdefinedtobe 1 2 |γ (s)||γ (t)| E(γ ,γ )= 1 2 ds dt. 1 2 | − |2 S1×S1 γ1(s) γ2(t) This energy was introduced by Freedman, He and Wang in [8] (see also O’Hara [24] for knot energies). It is again, like the Willmore energy of surfaces, conformally invariant. It was conjectured by Freedman-He-Wang (1994) [8] that the Möbius energy of any nontrivial link (one that cannot be isotoped to the union of the boundaries

20 FERNANDO C. MARQUES AND ANDRE´ NEVES of two disjoint disks) should be at least 2π2. The equality should be attained by the stereographic projection of the so-called standard Hopf link: 3 3 γˆ1(s)=(coss, sin s, 0, 0) ∈ S andγ ˆ2(t)=(0, 0, cos t, sin t) ∈ S .

By a result of He [13], it is enough to prove the conjecture for links (γ1,γ2)that have linking number lk(γ1,γ2)=±1. This is what I. Agol and the present authors prove in [1]:

1 3 3 Theorem. ([1]) Let γi : S → R , i =1, 2, be a 2-component link in R with 2 |lk(γ1,γ2)| =1.ThenE(γ1,γ2) ≥ 2π . 2 4 4 Moreover, if E(γ1,γ2)=2π then there exists a conformal map F : R → R such that (F ◦ γ1,F ◦ γ2) describes the standard Hopf link up to orientation.

The strategy is to construct a canonical family for the problem: if (γ1,γ2)in S3 ⊂ R4 denotes a 2-component link (note that the conformal invariance of the energy holds in any dimension [15]), we construct a 5-parameter family of surfaces in S3 with the same basic properties of the canonical family for the Willmore problem. Additionally, the area of any surface in the family is bounded above by the Möbius energy of the link. The family is constructed in such a way that if |lk(γ1,γ2)| = 1 then the center map Q : S3 → S3 associated with the family satisfies |deg(Q)| = 1. Therefore the 2π2 Theorem applies and we conclude the existence of at least one surface in the 2 2 family with area greater than or equal to 2π . Therefore E(γ1,γ2) ≥ 2π ,andafter some extra work one can also prove the rigidity statement. Let us sketch the construction of the family. First recall that the Gauss map 4 1 1 3 of (γ1,γ2)inR , denoted by g = G(γ1,γ2), is the Lipschitz map g : S × S → S defined by γ (s) − γ (t) g(s, t)= 1 2 . |γ1(s) − γ2(t)| We can always reparametrize the curves γ1,γ2 so that they are Lipschitz and parametrized proportionally to the arc length. Since | || | | | ≤ γ1(s) γ2(t) Jac g (s, t) 2 , |γ1(s) − γ2(t)| the parametrized torus g(S1 × S1) satisfies 1 1 area(g(S × S )) ≤ E(γ1,γ2). Given v ∈ R4,wedefinetheconformalmap x − v F : R4 \{v}→R4,F(x)= . v v |x − v|2 If v ∈ B4, we can compute 3 3 v Fv(S1 (0)) = S 1 (c(v)) where c(v)= 2 . 1−|v|2 1 −|v|

Hence, if gv denotes the Gauss map of the link (Fv ◦ γ1,Fv ◦ γ2), then it follows from the previous observation and the conformal invariance of the energy that 1 1 area(gv(S × S )) ≤ E(Fv ◦ γ1,Fv ◦ γ2)=E(γ1,γ2) for every v ∈ B4. Therefore we get a four-dimensional family of surfaces (parame- 3 trized tori) in S whose areas are bounded above by E(γ1,γ2). In order to apply the 2π2 Theorem, we need to introduce a ﬁfth parameter.

MIN-MAX THEORY OF MINIMAL SURFACES AND APPLICATIONS 21

4 4 Given w ∈ R and λ ∈ R,wesetDw,λ(x)=λ(x − w)+w,wherex ∈ R . 4 Finally, given v ∈ B and z ∈ (0, 1), we also define (2z − 1) b(v, z)= (1 −|v|2 + z)(1 − z) and (1 −|v|2)(2z − 1) a(v, z)=1+(1−|v|2)b(v, z)=1+ . (1 −|v|2 + z)(1 − z) For each v ∈ B4 fixed, z → a(v, z) is a nondecreasing parametrization of (0, +∞). 1 1 3 If γ1(S ) ∪ γ2(S ) ⊂ S , we define the canonical five-dimensional family of surfaces associated to (γ1,γ2)tobe 1 1 3 C(v, z)=g(v,z)(S × S ) ∈Z2(S ),

4 1 1 3 for (v, z) ∈ B × (0, 1), where g(v,z) : S × S → S is the Gauss map g(v,z) = G(Fv ◦ γ1,Dc(v),a(v,z) ◦ Fv ◦ γ2):

(Fv ◦ γ1)(s) − (Dc(v),a(v,z) ◦ Fv ◦ γ2)(t) g(v,z)(s, t)= . |(Fv ◦ γ1)(s) − (Dc(v),a(v,z) ◦ Fv ◦ γ2)(t)|

3 The surface C(v, z) must be interpreted as an element of Z2(S ). Both curves Fv ◦ γ1 and Dc(v),a(v,z) ◦ Fv ◦ γ2 are contained in spheres centered at c(v), and g(v,1/2) = G(Fv ◦ γ1,Fv ◦ γ2)giventhata(v, 1/2) = 1. 4 Thefactthatarea(C(v, z)) ≤ E(γ1,γ2) for every (v, z) ∈ B × (0, 1) follows from the conformal invariance of the energy and the estimates below: |(F ◦ γ )(s)||(D ◦ F ◦ γ )(t)| | | ≤ v 1 c(v),a(v,z) v 2 Jac g(v,z) (s, t) 2 |Fv ◦ γ1(s) − Dc(v),a(v,z) ◦ Fv ◦ γ2(t)| | ◦ || ◦ | ≤ a(v, z) (Fv γ1) (s) (Fv γ2) (t) 2 2 a(v, z)|Fv ◦ γ1(s) − Fv ◦ γ2(t)| + b(v, z) | ◦ || ◦ | ≤ (Fv γ1) (s) (Fv γ2) (t) 2 . |Fv ◦ γ1(s) − Fv ◦ γ2(t)|

4 3 The map C is uniformly continuous hence we get a map C : B ×[0, 1] →Z2(S ) with the following boundary behavior:

3.1. Proposition. There exists a constant c>0 such that for every p ∈ S3 we have

(i) C(p, 1/2) = −lk(γ1,γ2) · ∂Bπ/2(p), (ii) supp(C(p, z)) ⊂ Bπ/2(p) \ Br(z)(p) if z ∈ [1/2, 1], (iii) supp(C(p, z)) ⊂ Bπ/2(−p) \ Bπ−r(z)(−p) if z ∈ [0, 1/2], where b(z) 2z − 1 r(z)=cos−1 ∈ [0,π] and b(z)= . |b(z)|2 + c2 z(1 − z)

Remark. We have r(0) = π, r(1/2) = π/2, and r(1) = 0.

22 FERNANDO C. MARQUES AND ANDRE´ NEVES

The boundary values of C are not yet round spheres, but by the previous proposition they are contained in hemispheres varying continuously. The idea then is to apply area-decreasing deformations that take place inside these hemispheres and such that at the end of the deformations the surface is identiﬁed with a round sphere. After an appropriate reparametrization, we get:

3 3.2. Theorem. Let (γ1,γ2) be a 2-component link in S with lk(γ1,γ2)=−1. There exists a map 5 3 Φ:I →Z2(S ), obtained by reparametrizing an extension of C, that is continuous in the flat topology and that satisfies the following properties: (1) Φ(x, 0) = Φ(x, 1) = 0 for any x ∈ I4, 4 4 3 (2) Φ(x, t)=∂Br(t) (Q(x)) for every (x, t) ∈ ∂I × I,forsomeQ : ∂I → S , 4 (3) Φ(x, 1/2) = ∂Bπ/2(Q(x)) for any x ∈ ∂I , 5 (4) sup{area(Φ(x)) : x ∈ I }≤E(γ1,γ2), (5) the center map Q : ∂I4 → S3 satisfies deg(Q)=1.

4. Infinitely many minimal hypersurfaces Finally, we briefly describe an application of min-max theory to the problem of counting minimal hypersurfaces in Riemannian manifolds [21]. We say that a Riemannian manifold (M,g) satisfies the embedded Frankel property if any two smooth, closed, embedded minimal hypersurfaces of M intersect each other. We prove in [21]: Theorem 4.1. Let (M,g) be a compact Riemannian manifold of dimension (n +1),with2 ≤ n ≤ 6. Suppose that M satisfies the embedded Frankel property. Then M contains an infinite number of distinct smooth, closed, embedded, minimal hypersurfaces.

Since manifolds of positive Ricci curvature satisfy the embedded Frankel property [7], we derive the following corollary: Corollary 4.2. Let (M,g) be a compact Riemannian (n +1)-manifold with 2 ≤ n ≤ 6. If the Ricci curvature of g is positive, then M contains an inﬁnite number of distinct smooth, closed, embedded, minimal hypersurfaces.

The proof of Theorem 4.1 uses the Almgren-Pitts min-max theory for the area functional, combined with ideas from Lusternik-Schnirelmann theory. The idea is to apply min-max theory to the high-parameter families of hypersurfaces (mod 2 cycles) studied by Gromov [9–11]andGuth[12]. Let us give an informal overview of the proof. First note that the homotopy groups of the space of modulo 2 n-cycles in M, Zn(M,Z2), can be computed through the work of Almgren [2]. All homotopy groups vanish but the ∞ ﬁrst one: π1(Zn(M,Z2)) = Z2, just like for the topological space RP .Let 1 λ¯ ∈ H (Zn(M,Z2), Z2) be the generator. Gromov [9–11]andGuth[12] have studied continuous maps Φ from a simplicial ¯p ∗ ¯p ¯p complex X into Zn(M,Z2) that detect λ , in the sense that Φ (λ ) =0.Here λ denotes the p-th cup power of λ¯. An example can be given by starting with a Morse

MIN-MAX THEORY OF MINIMAL SURFACES AND APPLICATIONS 23 function f : M → R.Theopenset{x ∈ M : f(x)

Ψ(a0,...,ap)=∂ {x ∈ M : Pa(f(x)) < 0} .

Note that the open set {x ∈ M : Pa(f(x)) < 0} has ﬁnite perimeter, since − − { ∈ }⊂ 1 (1) ∪···∪ 1 (ka) x M : Pa(f(x)) = 0 f (ta ) f (ta )

(1) (ka) where ta ,...,ta are the zeros of Pa, ka ≤ p. The fact that we are using Z2 coeﬃcients implies that Ψ(a)=Ψ(−a), and therefore Ψ induces a map Φ : RPp → ∗ ¯p Zn(M; Z2). It satisﬁes Φ (λ ) =0. In fact, it follows from the results of Gromov and Guth that for every p ∈ N there exists a map Φ that detects λ¯p (with X = RPp)andsuchthat

1 sup area(Φ(x)) ≤ Cpn+1 , x∈RPp where C depends only on M. Guth’s construction [12]wasbasedonanelegant bend–and–cancel argument. ¯p If we denote by Pp the space of all maps that detect λ ,wehave(seealso [12, Appendix 3]):

1 ωp := inf sup area(Φ(x)) ≤ Cpn+1 , ∈P Φ p x∈dmn(Φ) where dmn(Φ) stands for the domain of Φ. We use Lusternik-Schnirelmann theory to show that if ωp = ωp+1 then there are infinitely many embedded minimal hypersurfaces. Theorem 4.1 is proven by contradiction, first assuming that there exist only finitely many smooth, closed, embedded minimal hypersurfaces. This implies that the sequence {ωp}p∈N is strictly increasing and, under the Frankel condition, each min-max volume ωp must be achieved by a connected, closed, embedded minimal hypersurface with some integer multiplicity. We use this to show that ωp must grow linearly in p and this contradicts the sublinear growth of ωp in p given by Gromov and Guth. There are many important questions related to min-max theory that are yet to be solved. Understanding the Morse index in relation to the number of parameters is one of them. Analyzing the area, the index, multiplicity and distribution of our minimal hypersurfaces are very interesting directions to pursue.

References [1] I. Agol, F. C. Marques and A. Neves, Min-max theory and the energy of links arXiv:1205.0825 [math.GT] (2012) 1–19. [2] Frederick Justin Almgren Jr., The homotopy groups of the integral cycle groups, Topology 1 (1962), 257–299. MR0146835 (26 #4355) [3] F. Almgren, The theory of varifolds. Mimeographed notes, Princeton (1965).

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[4] Matthias Bauer and Ernst Kuwert, Existence of minimizing Willmore surfaces of prescribed genus,Int.Math.Res.Not.10 (2003), 553–576, DOI 10.1155/S1073792803208072. MR1941840 (2003j:53086) [5] George D. Birkhoff, Dynamical systems with two degrees of freedom, Trans. Amer. Math. Soc. 18 (1917), no. 2, 199–300, DOI 10.2307/1988861. MR1501070 [6] W. Blaschke, Vorlesungen Uber¨ Differentialgeometrie III, Berlin: Springer (1929). [7] T. Frankel, On the fundamental group of a compact minimal submanifold, Ann. of Math. (2) 83 (1966), 68–73. MR0187183 (32 #4637) [8] Michael H. Freedman, Zheng-Xu He, and Zhenghan Wang, Möbius energy of knots and unknots,Ann.ofMath.(2)139 (1994), no. 1, 1–50, DOI 10.2307/2946626. MR1259363 (94j:58038) [9] M. Gromov, Dimension, nonlinear spectra and width, Geometric aspects of functional analysis (1986/87), Lecture Notes in Math., vol. 1317, Springer, Berlin, 1988, pp. 132–184, DOI 10.1007/BFb0081739. MR950979 (90d:58022) [10] M. Gromov, Isoperimetry of waists and concentration of maps, Geom. Funct. Anal. 13 (2003), no. 1, 178–215, DOI 10.1007/s000390300002. MR1978494 (2004m:53073) [11] Mikhail Gromov, Singularities, expanders and topology of maps. I. Homology versus volume in the spaces of cycles, Geom. Funct. Anal. 19 (2009), no. 3, 743–841, DOI 10.1007/s00039- 009-0021-7. MR2563769 (2012a:58062) [12] Larry Guth, Minimax problems related to cup powers and Steenrod squares,Geom. Funct. Anal. 18 (2009), no. 6, 1917–1987, DOI 10.1007/s00039-009-0710-2. MR2491695 (2010e:53071) [13] Zheng-Xu He, On the minimizers of the Möbius cross energy of links, Experiment. Math. 11 (2002), no. 2, 244–248. MR1959266 (2003k:58016) [14] Ernst Heintze and Hermann Karcher, A general comparison theorem with applications to volume estimates for submanifolds, Ann. Sci. Ecole´ Norm. Sup. (4) 11 (1978), no. 4, 451– 470. MR533065 (80i:53026) [15] Denise Kim and Rob Kusner, Torus knots extremizing the Möbius energy, Experiment. Math. 2 (1993), no. 1, 1–9. MR1246479 (94j:58039) [16] Rob Kusner, Estimates for the biharmonic energy on unbounded planar domains, and the existence of surfaces of every genus that minimize the squared-mean-curvature integral,El- liptic and parabolic methods in geometry (Minneapolis, MN, 1994), A K Peters, Wellesley, MA, 1996, pp. 67–72. MR1417949 (97k:53013) [17] Ernst Kuwert, Yuxiang Li, and Reiner Schätzle, The large genus limit of the infimum of the Willmore energy,Amer.J.Math.132 (2010), no. 1, 37–51, DOI 10.1353/ajm.0.0100. MR2597505 (2011c:58026) [18] H. Blaine Lawson Jr., Complete minimal surfaces in S3,Ann.ofMath.(2)92 (1970), 335– 374. MR0270280 (42 #5170) [19] Peter Li and Shing Tung Yau, A new conformal invariant and its applications to the Willmore conjecture and the first eigenvalue of compact surfaces, Invent. Math. 69 (1982), no. 2, 269– 291, DOI 10.1007/BF01399507. MR674407 (84f:53049) [20] Fernando C. Marques and André Neves, Min-max theory and the Willmore conjecture, Ann. of Math. (2) 179 (2014), no. 2, 683–782, DOI 10.4007/annals.2014.179.2.6. MR3152944 [21] Fernando C. Marques and André Neves, Min-max theory, Willmore conjecture and the energy of links, Bull. Braz. Math. Soc. (N.S.) 44 (2013), no. 4, 681–707, DOI 10.1007/s00574-013- 0030-x. MR3167128 [22] Henri Poincaré, Sur les lignes géodésiques des surfaces convexes (French), Trans. Amer. Math. Soc. 6 (1905), no. 3, 237–274, DOI 10.2307/1986219. MR1500710 [23] Jon T. Pitts, Existence and regularity of minimal surfaces on Riemannian manifolds,Mathe- matical Notes, vol. 27, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1981. MR626027 (83e:49079) [24] Jun O’Hara, Energy of a knot, Topology 30 (1991), no. 2, 241–247, DOI 10.1016/0040- 9383(91)90010-2. MR1098918 (92c:58017) [25] Antonio Ros, The Willmore conjecture in the real projective space, Math. Res. Lett. 6 (1999), no. 5-6, 487–493, DOI 10.4310/MRL.1999.v6.n5.a2. MR1739208 (2001a:53016) [26] Richard Schoen and Leon Simon, Regularity of stable minimal hypersurfaces, Comm. Pure Appl. Math. 34 (1981), no. 6, 741–797, DOI 10.1002/cpa.3160340603. MR634285 (82k:49054)

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[27] Leon Simon, Existence of surfaces minimizing the Willmore functional, Comm. Anal. Geom. 1 (1993), no. 2, 281–326. MR1243525 (94k:58028) [28] F. Smith, On the existence of embedded minimal 2-spheres in the 3–sphere, endowed with an arbitrary Riemannian metric, PhD thesis supervised by L. Simon, University of Melbourne (1982). [29] G. Thomsen, Uber¨ Konforme Geometrie, I: Grundlagen der Konformen Fl¨achentheorie, Abh. Math. Sem. Hamburg (1923), 31–56. [30] Francisco Urbano, Minimal surfaces with low index in the three-dimensional sphere,Proc. Amer. Math. Soc. 108 (1990), no. 4, 989–992, DOI 10.2307/2047957. MR1007516 (90h:53073) [31] T. J. Willmore, Note on embedded surfaces (English, with Romanian and Russian summaries), An. S¸ti. Univ. “Al. I. Cuza” Ia¸si Sect¸. I a Mat. (N.S.) 11B (1965), 493–496. MR0202066 (34 #1940) [32] Shing Tung Yau, Problem section, Seminar on Diﬀerential Geometry, Ann. of Math. Stud., vol. 102, Princeton Univ. Press, Princeton, N.J., 1982, pp. 669–706. MR645762 (83e:53029)

Princeton University, Department of Mathematics, Fine Hall, Princeton, NJ 08544 USA E-mail address: [email protected] Imperial College, Huxley Building, 180 Queen’s Gate, London SW7 2RH, United Kingdom E-mail address: [email protected]

Contemporary Mathematics Volume 656, 2016 http://dx.doi.org/10.1090/conm/656/13101

Homogenization on manifolds

Gonzalo Contreras

Abstract. We present a theorem by Contreras, Iturriaga and Siconolﬁ [8] in which we give a setting to generalize the homogenization of the Hamilton- Jacobi equation from tori to other manifolds.

A homogenization problem consists of a Partial Diﬀerential Equation (PDE) with a fast (oscillating) variable ε and a slow variable. The homogenization result is that when the oscillating period ε tends to zero, there is a limit of the solutions uε of the PDE to a solution of an homogenized or “averaged” PDE. An example of the homogenization result that we present here is the convergence of the average distance in the universal cover of the torus T2 = R2/Z2 to 2 2 the distance in the stable norm in H1(T , R)=R , when the diameter of the fundamental domain ε tends to zero (see ﬁg. 1).

Figure 1. Convergence to the stable norm.

In higher dimensions the minimal geodesics may not converge. This is related to the flats of the stable norm as in Hedlund’s example [14] in figure 2. Hedlund’s example is a 3-torus T3 = R3/Z3, in which the Riemannian metric is deformed in three disjoint tubes of different homological directions in which the

Partially supported by CONACYT, Mexico, grant 178838.

c 2016 American Mathematical Society 27

28 G. CONTRERAS

Figure 2. Hedlund’s example and its stable norm. central closed geodesics are very short. In the example, minimal geodesics follow the tubes with at most two jumps and the stable norm is (x, y, z) = |x| + |y| + |z|. In Hedlund’s example the minimal geodesics do not converge as ε → 0. There is a convergence as “holonomic measures” to an invariant measure supported on three periodic orbits on the tubes. The fact that there is no ergodic minimizing measure in a given homology class implies that the stable norm is flat on that class. An important observation in this geodesic example of homogenization is that the average minimal distance can be computed from the geodesics of the stable norm, which are straight lines. One expects that the homogenized or averaged problem is much simpler and computable than the original problem. Another application of homogenization theory is to obtain macroscopic laws from microscopic data. Homogenization theory has mostly been done in a periodic setting (i.e. on the torus Tn) or in quasi-periodic tilings or random media on Rn.Inthecaseof the Hamilton-Jacobi equation, the limiting objects are well known and naturally defined on arbitrary manifolds: the effective Lagrangian is Mather’s minimal action function β and the effective (or homogenized) Hamiltonian is its dual β∗,alsoknown as Mañé’s critical value. Nevertheless this homogenizations have only been made in Tn. We will show how to extend the homogenization result for the Hamilton-Jacobi equation from the torus Tn to an arbitrary compact manifold. We hope that the setting presented here can be applied to many other homogenization results.

1. Homogenization of the Hamilton-Jacobi equation Let M be a compact manifold without boundary. A Tonelli Lagrangian is a C2 function L : TM → R satisfying: ∂L ∀ ∈ (i) Convexity: ∂v ∂v (x, v) is positive deﬁnite (x, v) TM. L(x,v) ∞ ∈ (ii) Superlinearity: lim|v|→+∞ |v| =+ uniformly on x M. Examples of Tonelli Lagrangians are

HOMOGENIZATION ON MANIFOLDS 29

1 (1) The kinetic energy: L(x, v)= 2 v x, which gives the geodesic ﬂow and whose homogenization is equivalent to the examples given above. 1 − (2) The Mechanical Lagrangian: L(x, v)= 2 v x U(x) = kinetic energy - potential energy. This Lagrangian gives rise to Newton’s law with force F = −∇U(x). The action of a smooth curve γ :[0,T] → M is T AL(γ)= L γ(t), γ˙ (t) dt. 0 Critical points of AL satisfy the Euler-Lagrange equation d ∂L ∂L (1) = . dt ∂v ∂x The Euler-Lagrange equation is a second order equation whose solutions give rise to the Lagrangian Flow: ϕt : TM → TM,

ϕt(x, v)=(γ(t), γ˙ (t)), where γ is the solution of (1) with initial conditions (γ(0), γ˙ (0)) = (x, v). The convex dual of the Lagrangian is the Hamiltonian H : T ∗M → R H(x, p)= sup p(v) − L(x, v) . v∈TxM → ∗ ∂L The Legendre Transform Lv : TM T M, Lv(x, v)= ∂v (x, v), converts the Euler-Lagrange equation (1) into the Hamiltonian equations:

d Lv x˙ = Hp Lv = Lx ===⇒ dt p˙ = −Hx and conjugates the Lagrangian and Hamiltonian ﬂows. The Hamilton-Jacobi equation

(2) ∂tu + H(x, ∂xu)=0

encodes the minimal (Lagrangian) action cost. A solution u : M × R+ → R,tothe Hamilton-Jacobi equation with initial condition u(x, 0) = f(x) is given by the Lax formula t u(x, t) = inf f(γ(0)) + L(γ,γ˙ ) γ ∈ C1([0,t],M),γ(t)=x . 0 The characteristics of the Hamilton-Jacobi equation are Tonelli minimizers i.e. minimizers of the action with ﬁxed endpoints and ﬁxed time interval. The value of the solution is the initial value + the action along these minimizers. Tangent vectors to the characteristics are related to ∂xu through the Legendre Transform Lv:

(3) ∂xu = Lv(γ,γ˙ ). Usually there are no global classical solutions of the Hamilton-Jacobi equation due to crossing of characteristics as in ﬁgure 3. Indeed, from (3) at a crossing point there are various candidates for ∂xu, and hence ∂xu does not exist. There are two popular types of weak solutions in PDEs:

30 G. CONTRERAS

Figure 3. Crossing of characteristics.

• Weak solutions with weakly differentiable functions and Sobolev Spaces are inspired on the formula of integration by parts. • The viscosity solution is inspired on the maximum principle for PDEs. The first definition of viscosity solutions was made by L.C. Evans in 1980 [12]. Subsequently the definition and properties of the viscosity solutions of Hamilton- Jacobi equations were refined by Crandall, Evans and Lions in [9]. The existence and uniqueness of the viscosity solution of the initial value problem for the Hamilton-Jacobi equation was proved by Crandall and Lions in [10]. A continuous function is a viscosity solution of

∂tu + H(x, ∂xu)=0 1 if for every open set U ⊂ M and any φ ∈ C (U × R+, R):

• if u − φ attains a local maximum at (y0,t0) ∈ U × R+,then ∂tφ(y0,t0)+H(y0,∂x(y0,t0)) ≤ 0. • if u − φ attains a local minimum at (y0,t0) ∈ U × R+,then ∂tφ(y0,t0)+H(y0,∂x(y0,t0)) ≥ 0. 1.1. Theorem (Lions, Papanicolaou, Varadhan [15], Evans [13]). Let H : Rn × Rn → R be a Zn-periodic Tonelli Hamiltonian. For ε small let n fε : R → R be Lipschitz. Consider the Cauchy problem for the Hamilton-Jacobi equation ε x ε (4) ∂tu + H ε ,∂xu =0, ε u (x, 0) = fε(x). ε If limε fe = f uniformly then lim u = u uniformly, where u is the solution to

∂tu + H(∂xu)=0, u(x, 0) = f(x). The function H : Rn → R, called the eﬀective Hamiltonian is convex, superlinear and is independent of the variable x. The solutions to the homogenized problem can be easily written because the characteristics are straight lines and p = ∂ u is constant along them x p˙ = −Hx =0,

x˙ = Hp = constant.

HOMOGENIZATION ON MANIFOLDS 31

Thus − u(y, t)= min f(x)+t L y x , x∈Rn t where L(x, v)=max p(v) − H(p) p∈Rn is the Effective Lagrangian. It turns out that the Effective Lagrangian L = β is Mather’s minimal action n function β : H1(T , R) → R. The Effective Hamiltonian is related to Mañé’s critical value by H(P )=α(P )=c(L − P ),P∈ H1(Tn, R), here (L − P )(x, v):=L(x, v) − ωx(v), where ω is a closed 1-form in the cohomology class P . As such, it has several interpretations (see [7]): (i) α is the convex dual of β. (ii) α(P ) = inf k ∈ R | (L − P + k) ≥ 0 ∀ closed curve γ in Tn . γ (iii) α(P ) = inf k ∈ R | Φk > −∞ ,whereΦk : M × M → R is − | Tn Φk(x, y):=inf γ (L P + k) γ curve in from x to y , i.e. the minimal action with free time interval.1 (iv) α(P )=− inf (L − P ) dμ | μ is an invariant measure for L . (v) α(P ) is the energy level containing the support of the invariant measures μ which minimize (L − P ) dμ. (vi) α(P )= min max H(x, P + dxu). u∈C1(Tn,R) x∈Tn (vii) α(P ) is the minimum of the energy levels which contain a Lagrangian graph in T ∗Tn with cohomology class P . (viii) From Fathi’s weak KAM theory, α(P ) is the unique constant for which there are global viscosity solutions of the Hamilton-Jacobi equation n H(x, P + dxv)=α(P ),x∈ T . We explain briefly why Theorem 1.1 and (viii) imply that the Effective Hamil- tonian H is Mather’s alpha function α. Consider the case of affine initial conditions. The problem f(x)=u(x, 0) = a + P · x (5) ∂tu + α(∂xu)=0 has solution u(x, t)=a + P · x − α(P )t. n n Let v : T × R+ → R be a Z -periodic solution to the “cell problem”: n H(x, P + dxv)=α(P ),v: T × R+ → R. Let uε(x, t):=u(x, t)+εv x , ε ε x Fε(x):=u (x, 0) = f(x)+εv ε .

1 The function Φk is called Ma˜n´e’s action potential.

32 G. CONTRERAS

ε Then u solves ε x ε − x x ∂tu + H ε ,∂xu = α(P )+H ε ,P + ∂yv ε =0, ε u (x, 0) = fε(x). ε Also we have that fε → f and u → u uniformly and by (5) u satisﬁes a Hamilton- Jacobi equation with Hamiltonian α. Therefore Theorem 1.1 implies that H(P )= α(P ).

1.1. The Problems.

The generalization of Theorem 1.1 to other manifolds has three problems: x 1. It is not clear how to choose the generalization of ε . 2a. Equation (4) is the Hamilton-Jacobi equation for the Hamiltonian x Hε(x, p):=H( ε ,p), where p “remains the same”. Itisnotclearhow to do it in non-parallelizable manifolds where the parallel transport depends on the path. 2b. The effective Hamiltonian H(P ) “does not depend on x”. This is another version of the same problem 2a. 3. The candidate for effective Hamiltonian is Mather’s α function α : H1(M,R) → R. But in general dim H1(M,R) =dim M, i.e. the limit PDE would be in a space with different dimension, the differential structure would be destroyed. In fact, the Hamilton-Jacobi equation is an encoding of a variational principle (the minimal cost function) that will be stable under the change of space. The torus M = Tn has many coincidences that allow to formulate Theo- rem [15]: (1) Its universal cover satisfies

n n n 1 n T = R = H1(T , R)=H (T , R). The effective Hamiltonian H = α : H1(Tn, R)=Rn → R and the effective Lagrangian L = β : H1(M,R) → R are defined in the same space as the original periodic Hamiltonian. Thus the original PDE and the limit equation are in the same space. (2) The cotangent bundle is trivial: T∗Tn = Tn × Rn and the parallel transport does not depend on the path. Thus we can talk of a Hamiltonian that does not depend on x and the Hamilton-Jacobi equation for the effective Hamiltonian ∂tu + H(∂xu) = 0 makes sense.

1.2. The solution.

Problem 3. We start with the solution to problem 3: the space for the family of PDEs. Let M be a compact manifold without boundary and H : T ∗M → R a Tonelli Hamiltonian. Consider the Hurewicz homomorphism h : π1(M) → H1(M,R) which sends the homotopy class of a curve to its homology class with real coeﬃcients. The maximal free abelian cover M is the covering map M → M with group of Deck transformations k Deck(M)=Z =Im(h) ⊂ H1(M,R),

HOMOGENIZATION ON MANIFOLDS 33

where k =dimH1(M,R)andπ1(M)=kerh.

→ x Problem 1.[x ε ] Let d be the metric induced on M by the lift of the Riemannian metric on M. For problem 1 we use the metric spaces Mε := (M,εd). The maximal free abelian cover M has the structure of Zk, i.e. it is (perhaps a complicated) fundamental domain which is repeated as the points in Zk,asin k k ﬁgure 4. The space Mε has a “large scale structure” as εZ → R = H1(M,R).

Figure 4. The structure of M.

ε k k We think of Mε −−→ H1(M,R)asofεZ −→ R . For example: “linear maps on M” shall correspond to “integrals of closed 1-forms”. Our solutions of the ε-oscillation Hamilton-Jacobi equation will be uniformly Lipschitz on Mε, i.e. εK- ε ε k Lipschitz on Mε.SothatasolutionU on Mε will deﬁne a function rv on εZ which is K-Lipschitz. By an Arzel´a-Ascoli argument we will obtain a convergence ε k v → v on R = H1(M,R).

Figure 5. Example of a free abelian cover of a surface M = T2#T2 with group of Deck transformations Z3. It is not the maximal free abelian cover of M, because dim H1(M,R) = 4. The limit 3 space limε Mε = R has higher dimension than M.

Problem 2. [H independent of x] The solution to problem 2 consists on transforming the equation to an equivalent PDE. In the case of Rn as in problem (4)

34 G. CONTRERAS deﬁne vε : Rn × R+ → R by

ε ε x u (x, t)=:v ( e ,t). From (4) we obtain that vε is a solution to the problem ε 1 ε (6) ∂tv + H y, ε ∂yv =0, ε (7) v (y, 0) = fε(εy). Now equation (6) makes sense on any manifold. Equation (7) will make sense with the following deﬁnition of convergence of spaces.

1.3. Convergence of spaces.

This is inspired in Gromov’s Hausdorﬀ convergence but it is made ad hoc for our homogenization problem. We will only need quasi-isometries because since we are doing analysis, just the equivalence class of the norms matter. Let (M,d), (Mn,dn) be metric spaces and Fn :(Mn,dn) → (M,d) a continuous function. We say that limn(Mn,dn,Fn)=(M,d)if

(a) There are B,An > 0, with limn An =0suchthat −1 ∀x, y ∈ Mn : B dn(x, y) − An ≤ d Fn(x),Fn(y) ≤ Bdn(x, y).

(b) For all y ∈ M and n there are xn ∈ Mn with limn xn = y. Observe that (b) is a kind of surjectivity condition. And (a) implies that ∀ ∈ −1{ }≤ −−n→ y M : diam Fn y BAn 0, a kind of injectivity condition. If limn(Mn,dn,Fn)=(M,d), and fn(Mn,dn) → R, F (M,d) → R are continuous, we say that limn fn = f uniformly on compact sets if for every compact set K ⊂ M

lim sup |fn(x) − f(Fn(x))| =0. n −1 x∈Fn (K)

And we say that the family {fn} is equicontinuous if for every ε>0thereisδ>0 such that

∀n : x, y ∈ Mn,dn(x, y) <δ =⇒|fn(x) − fn(y)| <ε.

1 Fix a basis c1,...ck for H (M,R). Fix closed 1-forms ωi on M such that 1 ∗ ci =[ωi]. Deﬁne G : M → H1(M,R)=H (M,R) by x G(x) · ci = ω˜i , x0 whereω ˜i is the pullback of ωi on M.LetFε :(Mε,dε) → H1(M,R)beF (x):= εG(x). 1.2. Proposition. limε→0(M,εd,Fε)=H1(M,R)

HOMOGENIZATION ON MANIFOLDS 35

In the homogenized or averaged problem we will have that the (limit) positions are in the configuration space H1(M,R) and the momenta p and differentials ∂xu ∗ 1 R are in the dual of the configuration space H1 = H (M, ). This explains why the effective Lagrangian L = β : H1(M,R) → R is defined in the homology group H1(M,R) but the effective Hamiltonian is defined in the cohomology group H1(M,R). 1.3. Theorem (Contreras, Iturriaga, Siconolfi [8]). Let M be a closed Riemannian manifold. Let H : T ∗M → R be a Tonelli Hamiltonian and fε :(Mε,dε) → R continuous functions such that limε fε = f uniformly, with f : H1(M,R) → R Lipschitz. Let H be the lift of H to M and let vε be the solution to the problem ε 1 ε ∂tv + H y, ε ∂yv =0, ε v (y, 0) = fε(y).

ε Then the family v : Mε×]0, +∞[→ R is equicontinuous and ε lim v = u : H1(M,R) → R ε→0 uniformly on compact sets of H1(M,R)×]T0, +∞[, for any T0 > 0,whereu is the solution to

∂tu + H(∂xu)=0, u(x, 0) = f(x); and H : H1(M,R) → R is H = α Mather’s alpha function. 1.4. Subcovers. On abelian covers M with Deck transformation group D of the form D Zk ⊕ Z ⊕···⊕Z Z ⊕···⊕Z = a1 ap the limit limε(M,εd) will kill the torsion a1 ap , as in ﬁgure 6. Thus we may restrict to free abelian covers with group of Deck transformation without torsion D = Zk. These are sub covers of M.

Figure 6. The limit process kills the torsion: ε (Z4 ⊕ Z) → R.

Using equivariance properties of the Hamilton-Jacobi equation, we obtain as a corollary of Theorem 1.3 a similar result for other free abelian covers.

36 G. CONTRERAS

2 Figure 7. Z -cover of the surface M3 of genus 3. In this case 3 H1(M3, R)=R .

1.5. Speculations.

There are generalizations of Aubry-Mather theory which can be interpreted as a homogenization besides Tn or Zn and should give results in the setting presented above. On a generalization originated by Moser [17], Caffarelli, de la Llave and Valdinocci extend Aubry-Mather theory to higher dimensions on very general manifolds, see [11, remark 2.6], [5], [4]. There is also an extension by Candel and de la Llave [6] of the Aubry-Mather theory in statistical mechanics to configuration sets more general than Zn. Viterbo’s symplectic homogenization [19] has also been extended to general manifolds by Monzner, Vichery and Zapolsky [16]. Most of the homogenization theory is made only for the torus Tn.SomePDE’s techniques go through this setting despite the destruction of the differential structure in the limit. For example in the homogenization of the Hamilton-Jacobi equation, Evans perturbed test function method goes through to give a proof of the same result. The translation of homogenization results to manifolds can give interesting geometric objects. We have the following examples: • The homogenization of the geodesic flow gives the stable norm. The stable norm was used by Burago and Ivanov in their proof of the Hopf conjecture [2]. Bangert [1, Th. 6.1] proves that a metric on T2 whose stable norm is euclidean is the flat metric on T2. Osuna [18] proves that if Tn has the 1-dimensional and (n − 1)-dimensional stable norms Euclidean then the metric is flat. • The homogenization of the Hamilton-Jacobi equation gives Mather’s alpha function or Mañé’s critical value as the effective Hamiltonian. In this case the limiting object H(P ) was known independently of homogenization and had many interesting characterizations besides homogenization: variational, ergodic, geometric, symplectic as in (i)–(viii). Another example of a possible result is the homogenization of the Riemannian Laplacian. Let M be a closed manifold and Ω ⊂ H1(M,R)adomain.Letf : ∂Ω → 1 R and F :Ω→ R be continuous functions. Choose a basis [ωi]forH (M,R)and

HOMOGENIZATION ON MANIFOLDS 37

let Gε : Mε → H1(M,R)be x Gε(x) · [ωi]=ε ωi. x0

Figure 8. Homogenization of the Riemannian Laplacian.

Let vε be the solution to the problem ◦ −1 Δvε = F Gε on Gε (Ω), ◦ −1 vε = f Gε on ∂Gε (Ω).

Prove that vε → u where ∂2u A = F on Ω, ij ∂x ∂x ij i j u = f on ∂Ω. In this homogenized Laplacian we should have that

Aij = ηi(x),ηj (x) M ∗ where ·, · is the induced inner product in T M and ηi is the harmonic 1-form in the class [ωi]. Other questions can be: • Homogenization of the eigenvalue problem for the Riemannian Laplacian. • Probabilistic proofs of the homogenization of the Laplacian. • Homogenization of the discretization of the Laplacian on graphs. • Does it always give the same eﬀective Laplacian? Also for the wave and heat equations? • What about quasi-periodic arrays of manifolds? • What about non-abelian covers?

38 G. CONTRERAS

For non-abelian covers there is a forthcoming work by Alfonso Sorrentino. The Gromov-Hausdorﬀ tangent cone of the covering [3] should give the eﬀective space.

References [1] Victor Bangert, Geodesic rays, Busemann functions and monotone twist maps,Calc.Var. Partial Differential Equations 2 (1994), no. 1, 49–63, DOI 10.1007/BF01234315. MR1384394 (97b:53041) [2] D. Burago and S. Ivanov, Riemannian tori without conjugate points are flat, Geom. Funct. Anal. 4 (1994), no. 3, 259–269, DOI 10.1007/BF01896241. MR1274115 (95h:53049) [3] Dmitri Burago, Yuri Burago, and Sergei Ivanov, A course in metric geometry, Graduate Studies in Mathematics, vol. 33, American Mathematical Society, Providence, RI, 2001. MR1835418 (2002e:53053) [4] Luis Caffarelli, A homogenization method for non variational problems, Current developments in mathematics, 2004, Int. Press, Somerville, MA, 2006, pp. 73–93. MR2459291 (2009m:35021) [5] Luis A. Caffarelli and Rafael de la Llave, Planelike minimizers in periodic media, Comm. Pure Appl. Math. 54 (2001), no. 12, 1403–1441, DOI 10.1002/cpa.10008. MR1852978 (2002j:49003) [6] A. Candel and R. de la Llave, On the Aubry-Mather theory in statistical mechanics, Comm. Math. Phys. 192 (1998), no. 3, 649–669, DOI 10.1007/s002200050313. MR1620543 (99b:82018) [7] Gonzalo Contreras and Renato Iturriaga, Global minimizers of autonomous Lagrangians,22o Colóquio Brasileiro de Matemática. [22nd Brazilian Mathematics Colloquium], Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 1999. MR1720372 (2001j:37113) [8] Gonzalo Contreras, Renato Iturriaga, and Antonio Siconolfi, Homogenization on arbitrary manifolds, Calc. Var. Partial Differential Equations 52 (2015), no. 1-2, 237–252, DOI 10.1007/s00526-014-0710-4. MR3299180 [9] M. G. Crandall, L. C. Evans, and P.-L. Lions, Some properties of viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc. 282 (1984), no. 2, 487–502, DOI 10.2307/1999247. MR732102 (86a:35031) [10] Michael G. Crandall and Pierre-Louis Lions, On existence and uniqueness of solutions of Hamilton-Jacobi equations, Nonlinear Anal. 10 (1986), no. 4, 353–370, DOI 10.1016/0362- 546X(86)90133-1. MR836671 (87f:35052) [11] Rafael de la Llave and Enrico Valdinoci, A generalization of Aubry-Mather theory to partial differential equations and pseudo-differential equations, Ann. Inst. H. PoincaréAnal. Non Linéaire 26 (2009), no. 4, 1309–1344, DOI 10.1016/j.anihpc.2008.11.002. MR2542727 (2011d:37106) [12] Lawrence C. Evans, On solving certain nonlinear partial differential equations by accretive operator methods,IsraelJ.Math.36 (1980), no. 3-4, 225–247, DOI 10.1007/BF02762047. MR597451 (82b:35032) [13] Lawrence C. Evans, Periodic homogenisation of certain fully nonlinear partial differential equations, Proc. Roy. Soc. Edinburgh Sect. A 120 (1992), no. 3-4, 245–265, DOI 10.1017/S0308210500032121. MR1159184 (93a:35016) [14] Gustav A. Hedlund, Geodesics on a two-dimensional Riemannian manifold with periodic coefficients, Ann. of Math. (2) 33 (1932), no. 4, 719–739, DOI 10.2307/1968215. MR1503086 [15] P.-L. Lions, G. Papanicolau, and S. R. S. Varadhan, Homogenization of Hamilton-Jacobi equations, preprint, unpublished, 1987. [16] Alexandra Monzner, Nicolas Vichery, and Frol Zapolsky, Partial quasimorphisms and qua- sistates on cotangent bundles, and symplectic homogenization,J.Mod.Dyn.6 (2012), no. 2, 205–249, DOI 10.3934/jmd.2012.6.205. MR2968955 [17] Jürgen Moser, Minimal solutions of variational problems on a torus,Ann.Inst.H.Poincaré Anal. Non Linéaire 3 (1986), no. 3, 229–272. MR847308 (88a:58058) [18] Osvaldo Osuna, Rigidity of the stable norm on tori (English, with English and Spanish summaries), Rev. Colombiana Mat. 44 (2010), no. 1, 15–21. MR2733391 (2011j:53069) [19] Claude Viterbo, Symplectic homogenization, Preprint arXiv:0801.0206, 2007.

HOMOGENIZATION ON MANIFOLDS 39

CIMAT, A.P. 402, 36.000, Guanajuato. GTO, Mexico´ E-mail address: [email protected]

Contemporary Mathematics Volume 656, 2016 http://dx.doi.org/10.1090/conm/656/13102

Lagrangian cobordism: Rigidity and ﬂexibility aspects

Octav Cornea

Abstract. We survey recent work (Biran and Cornea (2013, 2014), Charette and Cornea (to appear in Israel J. Math.)) that relates Lagrangian cobordism to the triangulated structure of the derived Fukaya category as well as the background and a number of consequences.

Contents 1. Introduction 2. Background 3. Cobordism categories and the category T SDFuk∗(M) 4. The functorF and some of its properties 5. Sketch of the construction ofF References

1. Introduction The development of modern symplectic topology is articulated around the in- terplay of two seemingly opposite points of view: the first, “soft”, with roots in classical differential topology, centers on flexibility phenomena. That flexibility is present in symplectic geometry is easy to expect given that, by the Darboux and Weinstein theorems, local symplectic geometry is trivial. The second point of view, “hard”, originating in algebraic geometry and analysis, emphasizes rigidity. The rigid perspective is also natural but for a more subtle reason, namely the discovery by Gromov [16] that almost complex complex structures that are compatible with the symplectic form share many properties with true complex structures and, at the same time, are abundant. The dichotomy rigidity-flexibility is a useful perspective also in what concerns the topology of Lagrangian submanifolds that is our focus in this paper. There are two techniques that establish relations among Lagrangians: the first, originating in the flexible camp, is based on cobordism, a notion central to differential topology since the work of Thom in the ’50’s and introduced in the Lagrangian setting by Arnold [1]; the second, fundamentally rigid, originates in the work of Gromov and Floer [14] and is based on symplectic intersection theory.

The author was supported by an NSERC Discovery grant and a FQRNT Group Research grant.

c 2016 American Mathematical Society 41

42 OCTAV CORNEA

Given a symplectic manifold, (M 2n,ω), a typical output of the first technique is the cobordism group Gcob(M). As in the smooth case, Gcob(M)isdefinedasthe quotient of a free group generated by the Lagrangian submanifolds in M modulo relations given by Lagrangian cobordisms. The second, “rigid”, perspective also leads to a group, K0(DFuk(M)), the Grothedieck group of the derived Fukaya category of M. The derived Fukaya category DFuk(M) is a canonical triangulated completion of the Donaldson category of M, Don(M). Its detailed construction appears in Seidel’s book [32]. In turn, Don(M) has as objects the Lagrangian submanifolds L ⊂ M and as morphisms the Floer homology groups MorDon∗(M)(L, L )=HF(L, L ). The relation with intersection theory comes from the fact that HF(L, L) is the homology of a chain complex generated (generically) by the intersection points of L and L. For the Floer homology groups and the Fukaya categories etc to be defined, the Lagrangians involved have to be submitted to certain constraints. We denote by L∗(M) the appropriate class of Lagrangians. In this paper, this is a certain class of monotone Lagrangians, see §2.1 and §2.2 (see also Remark 5.1). We add an ∗ to the notation to indicate that all involved Lagrangians belong to this class. This F ∗ ∗ applies to K0(D uk (M)) as well as to Gcob(M)etc.

Once this constraint is imposed, the two groups are related by a surjective morphism [9]: ∗ → F ∗ (1) Θ : Gcob(M) K0(D uk (M)) . The existence of Θ follows from the fact that there is [9] a functor (2) F : Cob∗(M) → T SDFuk∗(M) relating a cobordism category Cob∗(M)andanenrichment,T S DFuk∗(M), of the derived Fukaya category DFuk∗(M). The morphism Θ can be viewed as a sort of an analogue of the classical Thom morphism relating smooth cobordism groups to the homotopy groups of certain universal spaces, now called Thom spaces.

The purpose of this paper is to review the the main properties of F and Θ and to survey the background. The main constructions are sketched and we provide some ideas of proofs. For more details we refer to [8–10].

2. Background 2.1. Basic definitions. We consider in this paper a fixed symplectic manifold (M 2n,ω) that is closed (or tame at infinity [4]). We recall that ω is a 2-form that n is closed and non-degenerate. A submanifold L ⊂ M is Lagrangian if ω|TL ≡ 0. Given such a Lagrangian L, there are two natural morphisms

μ : π2(M,L) −→ Z ,ω: π2(M,L) −→ R the ﬁrst called the Maslov index and the second given by integration of ω. We will also need another standard convention in the subject: we put

NL =inf{μ(α):α ∈ π2(M,L),ω(α) > 0} . This number is considered = ∞ if there is no class α with ω(α) > 0. A Lagrangian L is called monotone if there exists ρ>0 so that the two morphisms above are

LAGRANGIAN COBORDISM: RIGIDITY AND FLEXIBILITY ASPECTS 43 proportional with constant of proportionality ρ, ρμ(α)=ω(α), ∀ α ∈ π2(M,L)and NL ≥ 2. Remark 2.1. A simple way to think about monotonicity is as a form of symmetry. For instance, the sphere S2 with the standard volume form is a symplectic manifold and any equator (that is a circle that divides the sphere in two parts of equal area) is a monotone Lagrangian submanifold of S2.AcircleonS2 that is not an equator (in this sense) is not monotone. A particular class of monotone Lagrangians are exact ones. In this case the symplectic form ω admits a primitive, η, dη = ω and the 1-form η|L is itself exact. The number NL is = ∞ for exact Lagrangians L.

The next deﬁnition is a variant of a notion ﬁrst introduced by Arnold [1, 2].

2 2 2 Endow R with the symplectic structure ω0 = dx ∧ dy,(x, y) ∈ R and R × M 2 2 with the symplectic form ω0 ⊕ ω.Letπ : R × M → R be the projection. For a 2 2 −1 subset V ⊂ R × M and S ⊂ R we let V |S = V ∩ π (S). Definition . 2.2 Let (Li)1≤i≤k− and (Lj)1≤j≤k+ be two families of closed La- grangian submanifolds of M. We say that that these two (ordered) families are Lagrangian cobordant, (Li) (Lj), if there exists a smooth compact cobordism ⊂ × R × (V ; i Li, j Lj ) and a Lagrangian embedding V ([0, 1] ) M so that for some >0wehave: V |[0,)×R = ([0,) ×{i}) × Li i (3) | − ×{ } × V (1−,1]×R = ((1 , 1] j ) Lj . j The manifold V is called a Lagrangian cobordism from the Lagrangian family ; (Lj) to the family (Li). We denote such a cobordism by V :(Lj ) (Li)or (V ;(Li), (Lj)).

Figure 1. ; R2 A cobordism V :(Lj) (Li) projected on .

A cobordism is called monotone if V ⊂ ([0, 1] × R) × M is a monotone Lagrangian submanifold. We mostly view cobordisms as embedded in R2 × M. Given a cobordism V ⊂ ([0, 1] × R) × M as above we can extend trivially its negative ends towards −∞ and its positive ends to +∞ thus getting a Lagrangian V ⊂ R2 × M.Wedo

44 OCTAV CORNEA not distinguish between V and V .IfV ∈ C × M is a cobordism, then, outside a large enough compact set, V equals a union of its negative ends, of the form −∞ − ×{}× ∞ ×{}× ( , a] i Li, and its positive ends, of the form [a, ) i Li. There is also an associated notion of isotopy for cobordisms [8]: two cobordisms V,V ⊂ C × M are horizontally isotopic if there exists a hamitonian isotopy φt, t ∈ [0, 1] of C × M sending V to V and so that, outside of a compact, φt(V ) has the same ends as V for all t ∈ [0, 1] (in other words, the ends can slide along but their image in C × M - outside a large compact set - remains the same; the hamiltonian isotopy is not necessarily with compact support).

2.2. Rigidity: Floer theory and the Fukaya category. Starting from Gromov’s [16] breakthrough, rigidity properties are extracted from the behaviour of moduli spaces of J-holomorphic curves u :Σ→ M (see [24] for a modern, thorough treatment of the subject). Here Σ is a Riemann surface, in our case of genus 0, possibly with boundary. The almost complex structure J on M is compatible with the form ω (in the sense that ω(−,J−) is a Riemannian metric) and the! fact that u is J-holomorphic means du ◦ i = J ◦ du. In case Σ has boundary ∂Σ= Ci,then u maps the boundary components to Lagrangians Li ⊂ M, u(Ci) ⊂ Li.

In our setting, the ﬁrst important moduli space M(α, J) consists of J-ho- 2 1 lomorphic disks u :(D ,S ) → (M,L)sothat[u]=α ∈ π2(M,L) modulo reparametrizations of the domain. The notation means, in particular that u(S1) ⊂ L. Here, as above, L is a Lagrangian submanifold of M. The virtual dimension of this moduli space is = μ(α)+n − 3. If L is monotone and α is so that μ(α)=2, then, for generic J, this moduli space is a manifold of dimension n − 1, without boundary. The fact that there is no boundary follows from μ(α)=2andNL ≥ 2. Considering now the J-holomorphic disks u as before but together with one marked point P ∈ ∂D2 we obtain a moduli space M1(α, J)ofJ-holomorphic disks with one marked boundary point. It has dimension n and is again a manifold without boundary. This moduli space is endowed with an evaluation map ev : M1(α, J) → L, ev(u)=u(P ). Let dL = degZ2 (ev). It is easy to see, again due to the monotonicity condition, that dL is actually independent of J and is thus a simple enumerative invariant of L: itcounts(mod2)thenumberofJ-holomorphic disks through a generic point.

We now briefly describe the most fundamental tool in modern symplectic topology: Floer homology. In our context it is defined (following [14], [26, 27]) for two Lagrangian submanifolds L, L both monotone with the same monotonicity constant ρ and, additionally, so that dL = dL . We also suppose that the two inclusion morphisms π1(L) → π1(M), π1(L ) → π1(M) have a torsion image. We also assume that L and L intersect transversely and that they are both closed. The Floer complex CF(L, L; J)isgivenby CF(L, L ; J)=(Z2 , d) with the differential defined as follows. For two intersection points x, y ∈ L ∩ L consider the moduli space of J-holomorphic curves u : R × [0, 1] → M with u(R × {0}) ⊂ L and u(R ×{1}) ⊂ L , and that originate in x, lims→−∞u(s, t)=x,and arrive in y, lims→∞ u(s, t)=y. Such curves are called Floer strips. For generic J this moduli space, M(x, y; J), decomposes into connected components each of

LAGRANGIAN COBORDISM: RIGIDITY AND FLEXIBILITY ASPECTS 45

Figure 2. A Floer strip relating the intersection points x and y of L and L. which is a manifold. The dimensions of the different components is not necessarily M M the same, but, nevertheless, we put dx = y #( (x, y; J))y where #( (x, y; J)) is the count (mod 2) of the 0-dimensional components. It is a consequence of the Gromov compactness theorem, one of the keystones of the subject, that the sum before is finite and that d2 = 0. Further, the resulting Floer homology HF(L, L) is independent of J and is invariant with respect to Hamiltonian deformations of ∼ ∼ L and L in the sense that HF(L, L ) = HF(φ(L),L) = HF(L, φ(L )) where φ : M → M is a Hamiltonian isotopy. Additionally, if L is exact and φ(L)is ∼ transverse to L,thenHF(L, φ(L)) = H(L; Z2). Remark 2.3. a. Floer homology for monotone Lagrangians has been introduced by Oh [26]. Compared to the rather simplified setting discussed here a number of extensions are available. For instance, under additional assumptions there are variants that admit Z gradings and are defined over Z. There are also far reaching extensions beyond the monotone case [15]. b. The condition dL = dL is necessary for the following reason. By using the Gromov compactness theorem together with a gluing argument (gluing holomorphic disks to intersection points of L and L) one can show that the Floer “differential” d 2 verifies in general d x =(dL − dL )x. The condition on π1 (introduced in [26]) can be dropped by working over certain Novikov rings but, in the current formalism, where we count Floer trajectories directly over Z2, it is necessary to insure that the sums appearing in the Floer differential are finite. By viewing the strips that give the Floer differential as examples of polygons with punctures on the boundary - in this case with two sides and two punctures - one is easily led to more complicated moduli spaces and higher associated structures. These higher structures are assembled in the Fukaya A∞-category. We only sketch here the definition of this much richer structure and we refer to Seidel’s fundamental monograph [32] for details on the construction. First, we define more precisely the class of Lagrangians L∗(M) that we will work with: for this we fix ρ>0, d ∈ Z2. We denote the class of Lagrangians under consideration by Lρ,d(M). It consists of monotone Lagrangians L ⊂ M with monotonicity constant ρ and so that additionally:

(4) dL = d, π1(L) → π1(M) is null and HF(L, L) =0 .

As before, the condition on π1 is required to insure the ﬁniteness of certain algebraic sums. The condition HF(L, L) = 0 (which in the language of [7]meansthatL is not narrow) is imposed here because all the techniques described below basically do not “see” in any way those Lagrangians L so that HF(L, L)=0.Thus,in

46 OCTAV CORNEA essence, this condition gets rid of information that is irrelevant for our discussion. We also point out that there exists a meaningful deﬁnition of HF(L, L)eveniftwo Lagrangians L and L are not transversal, for instance when L = L. To shorten the notation we will continue to put L∗(M)=Lρ,d(M).

The first step is to construct the Donaldson category, Don∗(M). This is a category whose objects are the elements of L∗(M) and the morphisms are defined as Mor(L, L)=HF(L, L). The composition, also called the Donaldson triangle product, (5) ∗ : HF(L, L) ⊗ HF(L,L) → HF(L, L) is defined by using J-holomorphic polygons u : D2\{P, Q, R}→M with three 2 edges C1,C2,C3 that meet at the three punctures {P, Q, R}⊂∂D ,sothat∂C1 = {R, P },∂C2 = {P, Q},∂C3 = {Q, R}; further, the edges Ci are mapped to the Lagrangians L, L ,L as follows: u(C1) ⊂ L, u(C2) ⊂ L , u(C3) ⊂ L and, assymp- totically, the punctures go to intersection points of the Lagrangians involved.

Figure 3. A triangle contributing to the Donaldson product.

It is a non-trivial fact that this does indeed produce a product μ2 : CF(L, L) ⊗ CF(L,L) → CF(L, L) that is associative in homology. The lack of associativity at the chain level leads to the existence of higher operations: k μ : CF(L1,L2) ⊗ CF(L2,L3) ...⊗ CF(Lk,Lk+1) → CF(L1,Lk+1) that are defined using moduli spaces of polygons with k + 1 edges. For coherence of notation, we rename the Floer differential as μ1 : CF(L, L) → CF(L, L). With appropriate choices of auxiliary data - alsmost complex structures, Hamiltonian perturbations etc (technically these are quite complicated - see [32]) the μk’s satisfy relations of the type: (6) μi(−, −,...−,μj , −,...,−)=0. i+j=m ∗ k In other words, the objects in L (M) together with the operations μ form an A∞- category called the Fukaya category Fuk∗(M). While it is very difficult to work ∗ directly with Fuk (M), one can use this A∞ category to construct a triangulated

LAGRANGIAN COBORDISM: RIGIDITY AND FLEXIBILITY ASPECTS 47 completion of Don∗(M). Roughly, the construction is as follows. There exists a notion of module over an A∞ category. Specializing to our case, such a module M associates to each object L ∈L∗(M) a chain complex M(L) and higher operations k μM : CF(L1,L2) ⊗ ...⊗ CF(Lk−1 ⊗ Lk) ⊗M(Lk) →M(L1) that satisfy relations similar to (6). It is easy to define morphisms φ : M→M. They consist of chain morphisms φL : M(L) →M(L) together with appropriate higher components for each L ∈L∗(M). As a consequence, modules form them- ∗ selves an A∞-category, Mod (M). There is a functor (7) Y : Fuk∗(M) →Mod∗(M) called the Yoneda functor that is basically an inclusion and sends each object N ∈ ∗ L (M) to its associated Yoneda module defined by MN (L)=CF(L, N)(and appropriate higher operations). Given a morphism φ : M→M it is possible to construct the cone over it, C(φ). This is a module so that on each object L it coincides with the cone - in the category of chain complexes - over the chain map φL. Any sequence quasi- φ isomorphic to the sequence N −→ N → C(φ) is called exact. With this preparation, the derived Fukaya category DFuk∗(M) is obtained from Fuk∗(M) in two steps: first, we complete, inside Mod∗(M), the image of the Yoneda functor with respect to exact sequences thus getting a new A∞ category Fuk∗(M)∧; secondly, we put DFuk∗(M)=H(Fuk∗(M)∧). In other words DFuk∗(M) has the same objects as Fuk∗(M)∧ but its morphisms are the homological images of the morphisms in Fuk∗(M)∧.

The key property of DFuk∗(M) is that it is triangulated, with the exact triangles being the image of the exact triangles from Fuk∗(M)∧. Clearly, the Donaldson category is contained in DFuk∗(M), however the latter category contains, apri- ori, many more objects than the former. Basically, richer are the morphisms in Fuk∗(M), more objects are added to those in Don∗(M). As DFuk∗(M) is triangulated, it is possible to decompose objects L ∈L∗(M) ∗ with respect to others L1, L2 ...∈L (M). In the presence of such a decomposition one can recover properties of L from those of the Li’s. At the same time, one of the difficulties with this construction comes from the rather algebraic description of the exact triangles in DFuk∗(M) which makes them hard to detect in practice. We now use the triangulated structure of DFuk∗(M)toassociatetoitthe Grothendieck group ∗ K0(DFuk (M)) which is - in our non-oriented and ungraded case - the Z2-vector space generated by the objects of DFuk∗(M) modulo the relations M + M = M whenever M → M → M is an exact sequence. 2.3. Flexibility: h-principle and surgery. Most of the flexibility phenomena in symplectic topology are based on Gromov’s h(omotopy)-principle (see [17],[13],[23]). The particular application of the h-principle that is relevant for us here concerns Lagrangian immersions, see [4] for this form: (H) There is a weak homotopy equivalence between the space of Lagrangian immersions L → M and the space of bundle maps Φ : TL → TM that map each fibre TxL to a Lagrangian subspace of TxM and are so that the

48 OCTAV CORNEA

map φ : L → M, induced on the base, satisﬁes [φ∗ω]=0∈ H2(L; R). In particular, deciding whether a map f : L → M is homotopic to a Lagrangian immersion f : L → M reduces to an algebraic-topological veriﬁcation.

We also need an additional “ﬂexible” construction which is called Lagrangian surgery (see [21], [29]). We start by describing the local picture. Fix the following two Lagrangians: n n n n L1 = R ⊂ C and L2 = iR ⊂ C and consider the curve H ⊂ C, H(t)= a(t)+ib(t), t ∈ R, with the following properties (see also Figure 4): - H is smooth. -(a(t),b(t)) = (t, 0) for t ∈ (−∞, −1]. -(a(t),b(t)) = (0,t)fort ∈ [1, +∞). - a(t),b(t) > 0fort ∈ (−1, 1).

Figure 4. The curve H ⊂ C.

Let | ∈ R 2 ⊂ Cn L = (a(t)+ib(t))x1,...,(a(t)+ib(t))xn t , xi =1 . It is easy to see that L as defined above is Lagrangian. We will denote it by L = L1#L2 (with an abuse of nation as we omitted the handle). Moreover, it is also not difficult to construct [8] a cobordism V : L ; (L1,L2). In case L1 and L2 intersect in a single point, then L is diffeomorphic to the connected sum of L1 and L2 andonecansee(asin[8]) that the cobordism V above is homotopy equivalent to the wedge L1 ∨ L2.

By using the Weinstein neighbourhood theorem, the local picture can be implemented globally without diﬃculty. A few consequences of this construction are relevant here: (S1) If L ⊂ M is an immersed Lagrangian with transversal double points, then by surgery at each double point of L we obtain an embedded Lagrangian L ⊂ M. ; (S1) Similarly to the ﬁrst point: if V :(Li) (Lj ) is an immersed Lagrangian cobordism with transversal double points but so that the Li’s and the

LAGRANGIAN COBORDISM: RIGIDITY AND FLEXIBILITY ASPECTS 49

Lj ’s are embedded (same deﬁnition as in 2.2 but V is immersed, not necessarily embedded), then by surgery at the double points of V ,we ; obtain an embedded cobordism V :(Li) (Lj). ρ,d (S3) if L1,L2 ∈L (M) intersect in a single point, then L = L1#L2 ∈ ρ,d ρ,d L (M)iscobordantto(L1,L2) by a cobordism V so that V ∈L (C × M). To verify the last condition we use the cobordism constructed in [8] so that, as mentioned above, V L1 ∨ L2.GiventhatL1 and L2 intersect in a single point, this leads to a simple description of the group π2(C × M,V ) and as the monotonicity constant ρ is the same for both L1 and L2 we deduce that V is also monotone with the same monotonicity

constant. Interestingly, as we shall discuss later, dL1 = dL2 = d is not required here. By the notation V ∈L∗(C × M)wemeanthatV is monotone with respective constants (ρ, d)andthatπ1(V ) → π1(M) is trivial. There is no Floer homology condition imposed to V (this is in contrast to (4)).

We now define the Lagrangian cobordism groups associated to M. The simplest such cobordism group, Gcob(M), is defined as the free group generated by all closed, connected Lagrangian submanifolds L ⊂ M modulo the relations given by L1 · L2 · ...· Lk = 1 if there is a cobordism V : ∅ ; (L1,L2,...,Lk). There are, of course, many variants of this definition but the one of main G∗ interest to us is the monotone cobordism group, cob(M), which defined by first ∗ fixing ∗ =(ρ, d) and using the same definition as above but now with Li ∈L (M), V ∈L∗(C × M). It is also useful to consider the abelianizations of these groups Gcob(M) and, ∗ respectively, Gcob(M). Remark 2.4. i. Because we work in a non-oriented setting the two ∗ Z groups Gcob(M)andGcob(M) are actually 2-vector spaces. Moreover, it is easy to see that Gcob(M) is actually abelian so that Gcob(M)=Gcob(M). Indeed, consider two curves γ1,2 and γ2,1 in the plane so that they are both horizontal at ±∞ and so that γ1,2 is constant equal to 1 at +∞ and constant equal to 2 at −∞ while γ2,1 is constant to 2 at +∞ and equal to 1 at −∞. We assume that the two curves intersect transversely in one point. For any two Lagrangians L1, L2 we then define V :(L1,L2) → (L2,L1) by V = γ1,2 × L1 ∪ γ2,1 × L2.ThisV is obviously not embedded (except if L1 and L2 are disjoint) but by a small perturbation we may assume that it is immersed with only double points and then, as explained above, we can surger the double points and get an embedded cobordism V :(L1,L2) → (L2,L1)sothatL1 and L2 commute in Gcob(M). Notice ∗ also that if L1,L2 ∈L (M)andL1 and L2 are either disjoint or intersect in a single point, then - again, by the surgery argument - they commute G∗ in cob(M). ii. There are clearly even more refined variants of these cobordism groups that take into account orientations and possibly spin structures etc.

The property (S2) together with the h-principle for Lagrangian immersions as stated at (H) above imply that general cobordism is quite ﬂexible and that the “general” cobordism groups can be computed by algebraic-topological methods:

50 OCTAV CORNEA essentially, one uses the h-principle to compute a group deﬁned as above but by using immersed Lagrangians V and not embedded ones; one then shows, by the point (S2), that this group coincides with Gcob(M). Such calculations have been pursued by Eliashberg [12] and Audin [3].

3. Cobordism categories and the category T SDFuk∗(M) 3.1. Cobordism categories. The modern perspective on cobordism treats manifolds as objects in a category and the cobordisms relating them as morphisms in an appropriate category. This point of view is quite useful in our setting (see also [25] for an alternative approach).

The category of main interest for us here is Cob∗(M)(see[9] where it is denoted C d C ∗ ≥ by ob0(M)). The objects of ob (M) are families (L1,L2,...,Lr) with r 1, ∗ Li ∈L (M). Given two such families (L1,L2,...,Lr)and(K1,...Ks) a morphism

W :(K1,...,Ks) → (L1,L2,...,Lr) is an ordered family (W1,...,Ws)whereeachWi is a horizontal isotopy class ∈L∗ C × ; → of a cobordism Vi ( M)sothatV1 : K1 (L1,...Li1 ), V2 : K2 ; (Li1+1,...,Li2 ) ,..., Vs : Ks (Lis ,...,Lr) (for a more precise description see [9]). In particular, each of the Vj ’s has a single positive end that coincides with Kj . It is easy to see how to embedd the union (V1 ∪ ... ∪ Vs)asaLa- grangian in C × M so that it provides a cobordism (K1,...Ks) ; (L1,L2,...,Lr) and W can be viewed as the horizontal isotopy class of this cobordism. At the same time, notice that the horizontal isotopy class of an arbitrary cobordism U :(K1,...Ks) ; (L1,L2,...,Lr) is not in general a morphism in our category (for instance if U is connected and K1,K2 = ∅). Intuitively, a good way to view a basic morphism in our category:

V : K ; (L1,...,Li) is as a “formula” that decomposes the Lagrangian K into the pieces L1,...,Li. The composition of morphisms is induced by concatenation from right to left: V #V is obtained by gluing the negative ends of V to the positive ends of V . Remark 3.1. The reason why concatenation does not leave the class L∗(M)is precisely that each morphism is a union of cobordisms with a single positive end. With a little more care in deﬁning all of this, it is easy to see that Cob∗(M)has the structure of a monoidal category so that the operation on objects is given by

(L1,...,Lr), (K1,...,Ks) → (L1,...,Lr,K1,...,Ks) and similarly for morphisms.

We will also use another category that is a simpler version of Cob∗(M)andis denoted by SCob∗(M). Its objects are Lagrangians L ∈L∗(M) and its morphisms L → L are horizontal isotopy classes of cobordisms V : L ; (L1,...,Li,L), V ∈L∗(M). In other words, a morphism from L to L is represented by a cobordism with a single positive end that coincides with L and with possibly many negative ends but so that the “last” negative end is L. Composition is again induced by concatenation: if V : L → (K1,...,Kr,L ) represents a second morphism

LAGRANGIAN COBORDISM: RIGIDITY AND FLEXIBILITY ASPECTS 51

L → L, then the composition L → L → L is represented by the cobordism V #V : L ; (L1,...,Li,K1,...,Kr) deﬁned by gluing V to V along L and extending the ends L1,...,Li trivially in the negative direction. There is a functor P : Cob∗(M) → SCob∗(M) that is deﬁned at the level of objects by (L1,...,Lk) → Lk and similarly for morphisms.

3.2. Cone-decompositions in the derived Fukaya category. The purpose of the paper is to explain how the cobordism perspective on Lagrangian submanifolds, as reﬂected in the categories Cob∗(M)andSCob∗(M), is related to to the “rigid” invariants encoded in the derived Fukaya category, DFuk∗(M). There is however an immediate obstacle: the most important structural property of DFuk∗(M) is that it is triangulated while neither one of Cob∗(M)andSCob∗(M) are so, with the consequence that a functor from one of the cobordism categories to DFuk∗(M) will neglect precisely this triangulated structure. This is the issue that we deal with here, following [9]. Namely, we describe brieﬂy a rather formal construction that shows how to extract, out of a triangulated category, C, another category T SC whose morphisms parametrize the various ways to decompose an object by iterated exact triangles in C. We apply this construction to DFukd(M) thus getting the category T SDFuk∗(M) that is the target of the functor F from (2).

We recall [35] that a triangulated category C is an additive category together with a translation automorphism T : C→Cand a class of triangles called exact triangles T −1X −→u X −→v Y −→w Z that satisfy a number of axioms due to Verdier and to Puppe (see e.g. [35]). A cone decomposition of length k of an object A ∈Cis a sequence of exact triangles: −1 ui vi wi T Xi −→ Yi −→ Yi+1 −→ Xi ∼ with 1 ≤ i ≤ k, Yk+1 = A, Y1 = 0. (Note that Y2 = X1.) Thus A is obtained in k steps from Y1 = 0. To such a cone decomposition we associate the family l(A)=(X1,X2,...,Xk) and we call it the linearization of the cone decomposition. This definition is an abstract form of the familiar iterated cone construction in case C is the homotopy category of chain complexes. In that case T is the suspension functor TX = X[−1] and the cone decomposition simply means that each chain complex Yi+1 is obtained from Yi as the mapping cone of a morphism coming from ui some chain complex, in other words Yi+1 =cone(Xi[1] −→ Yi) for every i,and Y1 =0,Yk+1 = A. There is also a rather obvious equivalence relation among cone-decompositions. We will now define the category T SC called the category of (stable) triangle (or cone) resolutions over C. The objects in this category are finite, ordered families (x1,x2,...,xk) of objects xi ∈Ob(C). We will first define the morphisms in T SC with domain being a family formed by a single object x ∈Ob(C)andtarget(y1,...,yq), yi ∈Ob(C). For this, consider triples (φ, a, η), where a ∈Ob(C), φ : x → T sa is an isomorphism (in C) for some index s and η is a cone decomposition of the object a with linearization

52 OCTAV CORNEA

s1 s2 sq−1 (T y1,T y2,...,T yq−1,yq) for some family of indices s1,...,sq−1.Amor- phism Ψ : x −→ (y1,...,yq) is an equivalence class of triples (φ, a, η)asbefore up to a natural equivalence relation. We now deﬁne the morphisms between two general objects. A morphism

Φ ∈ MorT S C((x1,...xm), (y1,...,yn)) is a sum Φ = Ψ1 ⊕···⊕Ψm where Ψj ∈ MorT S C(xj, (yα(j),...,yα(j)+ν(j))), and α(1) = 1, α(j +1)=α(j)+ν(j)+1, α(m)+ν(m)=n.Thesum⊕ means here the obvious concatenation of morphisms. With this definition this category is strict monoidal, the unit element being given by the void family. See again [9]formore details as well as for the definition of the composition of morphisms (basically, this comes down to the refinement of cone resolutions). There is a projection functor (8) P : T SC−→ΣC Here ΣC stands for the stabilization category of C:ΣC has the same objects as C and the morphisms in ΣC from a to b ∈Ob(C) are morphisms in C of the form s a → T b for some integer s. The definition of P is as follows: P(x1,...xk)=xk and on morphisms it associates to Φ ∈ MorT S C(x, (x1,...,xk)), Φ = (φ, a, η), the composition:

φ s wk s P(Φ) : x −→ T a −→ T xk with wk : a → xk deﬁned by the last exact triangle in the cone decomposition η of a, −1 wk T xk −→ ak −→ a −→ xk .

In this paper we take C = DFuk∗(M). We will work here in an ungraded and non-oriented setting so that T = id and all the indexes si above equal 1.

4. The functor F and some of its properties 4.1. The main theorem and a few corollaries. With the preparation of the last section we can now state the main result surveyed in this paper. Theorem 4.1. [9] There exists a monoidal functor, F : Cob∗(M) −→ T S DFuk∗(M), with the property that F(L)=L for every Lagrangian submanifold L ∈L∗(M). In the remainder of this section we “unwrap” this statement and discuss its consequences.

Corollary 4.2. If V : L ; (L1,...,Lk) is a Lagrangian cobordism, then ∗ there exist k objects Z1,...,Zk in DFuk (M) with Z1 = L1 and Zk L which ﬁt into k − 1 exact triangles as follows:

Li → Zi−1 → Zi ∀ 2 ≤ i ≤ k. In particular, L belongs to the triangulated subcategory of DFuk∗(M) generated by L1,L2,...,Lk.

LAGRANGIAN COBORDISM: RIGIDITY AND FLEXIBILITY ASPECTS 53

This follows directly from Theorem 4.1: given that V represents a morphism in Cob(M) and in view of the deﬁnition of T S (−), the sequence of exact triangles in the statement is provided by F(V ).

There exists a simpliﬁed version

F : SCob∗(M) → DFuk∗(M) of F that can be made explicit easily. At the level of objects F(L)=L for each ∗ L ∈L (M). Concerning morphisms, for each cobordism V : L → (L1,...,Lk−1,L) that represents a morphism φ in SCcob∗(M) we deﬁne

F([V ]) ∈ homDFuk(L, L )=HF(L, L ) to be the image of the unity in HF(L, L) (induced by the fundamental class of L) through a morphism

(9) φV : HF(L, L) → HF(L, L ) , F([V ]) = φV ([L]) .

2 In turn, φV is given by counting Floer strips u : R×[0, 1] → R ×M with boundary conditions u(R ×{0}) ⊂ γ × L, u(R ×{1}) ⊂ V ,whereγ ⊂ R2, V are as in Figure 5 (with L = Lk).

Figure 5. A cobordism V ⊂ R2 × M with a positive end L and with L = Lk together with the projection of the J-holomorphic strips that deﬁne the morphism φV .

The fact that F determines F results from the commutativity of the diagram (10) which is itself a simple consequence of the construction of F.

F Cob∗(M) /T SDFuk∗(M)

(10) P P F SCob∗(M) /DFuk∗(M)

The functor F is particularly useful to state another simple consequence of Theorem 4.1.

54 OCTAV CORNEA

Corollary 4.3. Consider the Lagrangian cobordism V : L ; (L1,L2).If ∗ ∗ ∗ L, L1,L2 ∈L (M) and V ∈L (C×M), then there is an exact triangle in D Fuk(M) L Nf 2 NNN NNFN(V ) NNN NNN (11) F(V ) pp8L ppp ppp pp ppp F(V ) L1 where V and V are the cobordisms obtained by bending the ends of V as in Figure 6 below.

Figure 6. The cobordisms V and V , V obtained by bending the ends of V as indicated.

To unwrap the meaning of F further, ﬁx N ∈L∗(M). Consider the functor

∗ hom(N,−) hN : DFuk (M) −−−−−−→ (V, ×) where (V, ×) is the monoidal category of ungraded vector spaces over Z2,withthe monoidal structure × being direct product. We put HFN = hN ◦F so that we have the commutative diagram (12). F SCob∗(M) /DFuk∗(M) OOO OOO (12) OO hom(N,−) H OO FN O' (V, ×)

The functor HFN exhibits Floer homology HF(N,−) as a vector space valued functor deﬁned on a cobordism category. Here are some properties of HFN that follow easily from Theorem 4.1. Corollary 4.4. For any N ∈Lthe Floer homology functor

HFN : SCob(M) → (V, ×) deﬁned above veriﬁes:

LAGRANGIAN COBORDISM: RIGIDITY AND FLEXIBILITY ASPECTS 55

∗ i. For each L ∈L (M), HFN (L)=HF(N,L).IfV : L ; (L1,...,Lk−1,L) ∗ represents a morphism in SCcob (M),thenHFN ([V ]) is the morphism (−) ∗ φV ([L])) : HF(N,L) → HF(N,L )

given by the Donaldson product ( 5) with the element φV ([L]) where φV is as in ( 9). ii. If V has just two negative ends L1, L2 and V , V are as in Corollary 4.3, then there is a long exact sequence that only depends on the horizontal isotopy type of V

HFN (V ) HFN (V ) HFN (V ) ...−→ H FN (L2) −−−−−−→HFN (L1) −−−−−−→HFN (L) −−−−−→ H FN (L2) −→ ...

and this long exact sequence is natural in N.Inparticular,φV ([L]) ∗ φV ([L2]) = 0 and, similarly, φV (L2]) ∗ φV ([L1]) = 0. iii. More generally, if V has negative ends L1,L2,...,Lk with k ≥ 2,then there exists a spectral sequence EN (V ) so that: a. the E2 term of the spectral sequence satisﬁes:

(EN (V ))2 = ⊕iHFN (Li)

b. from E2 on, the terms of the spectral sequence only depend on the horizontal isotopy type of V . c. EN (V ) converges to HFN (L) and is again natural in N.

To end the section notice that Corrolary 4.2 and the deﬁnition of K0(−) directly ∗ ∗ imply that the mapping L (M) → K0(DFuk (M)) given by L → L induces an epimorphism ∗ → F ∗ Θ:Gcob(M) K0D uk (M) as stated in equation (1). Recent results of Haug [18] show that a version of Θ (de- ﬁned for a suitable class L∗(−)) and for M = T2 is an isomorphism. Interestingly, his proof makes use of homological mirror symmetry for the elliptic curve.

4.2. Further related properties. 4.2.1. Lagrangian suspension and Seidel’s representation. We begin by recalling two important constructions in symplectic topology. ∗ The first one is Seidel’s representation S : π1(Ham(M)) → QH(M) of the Hamiltonian diffeomorphism group with values in the invertible elements of the quantum homology of the ambient manifold [31]. There also exists a Lagrangian version of Seidel’s representation ([19],[20],[22]). As noticed in [10], after convenient “categorification”, this version of Seidel’s representation can be viewed as an action of the fundamental groupoid Π(Ham(M)) on DFuk∗(M). This action induces an action of Π(Ham(M)) on T SDFuk∗(M): (13) S :Π(Ham(M)) × T SDFuk∗(M) → T SFuk∗(M) . The second construction is Lagrangian suspension [30]. This too gives rise [10]to an action of Π(Ham(M)), this time on Cob∗(M), (14) Σ : Π(Ham(M)) ×Cob∗(M) →Cob∗(M) . It turns out that these two actions are interchanged by F. In fact, we have the following commutative diagram that “categorifies” Seidel’s representation:

56 OCTAV CORNEA

Theorem 4.5. [10] The following diagram of categories and functors com- mutes: S / ∗ π1(Ham(M)) QH(M)

i ∗ S / ∗ (15) Π(Ham(M)) End(T SDFuk (M))

Σ F∗ End(Cob∗(M)) /fun(Cob∗(M),TSDFuk∗(M)) F∗ The categories and functors in the top square are strict monoidal as is the functor Σ. Here the functor S is Seidel’s representation [31] viewed as a monoidal functor and the action ∗ is a refinement of the module action of quantum homology on ∗ Lagrangian Floer homology [7]. The functors F∗ and F are induced respectively by composition and pre-composition with F. Recall that an action M×C→C of a monoidal category M on a category C can be viewed as a strict monoidal functor M→End(C, C) and thus the commutativity of the bottom square in (15) means that F is equivariant with respect to S from (13) and Σ as in (14). A good part of the geometric content in Theorem 4.5 is reflected in the following particular case. Assume V is obtained by Lagrangian suspension with respect to a loop of Hamiltonian diffeomorphisms, g = {gt}, g0 = g1 = id. This means that we consider a time dependent Hamiltonian G : R × M → R that generates g (so that | | G ⊂ R × R × G is null for t large) and we put V =(t, G(t, x),φt (x)) M.Inthis case, the class φV ([L]), with φV from (9), coincides with S([g]) ∗ [L]where∗ is the module action ∗ : QH(M) ⊗ HF(L, L) → HF(L, L). 4.2.2. Lagrangian quantum homology. Let L ⊂ M be a montone Lagrnagian. −1 Denote by Λ = Z2[t ,t] the ring of Laurent polynomials in t, graded so that |t| = −NL.(IncaseL is weakly exact, i.e. ω(A) = 0 for every A ∈ π2(M,L)we put Λ = K.) The Lagrangian quantum homology QH(L) is the homology of a complex, C(D), called the pearl complex (see [5–7] for details). It is associated to a triple of auxiliary structures D =(f,(·, ·),J)wheref : L −→ R is a Morse function on L,(·, ·) is a Riemannian metric on L and J is an ω-compatible almost complex structure on M. With these structures fixed we have

C(D)=Z2Crit(f)⊗Λ and the differential of this complex counts so called pearly trajectories that consist of negative gradient flow lines of f with a finite number of points “replaced” with non-constant J-holomorphic disks as in Figure 7. The pearl complex is Z-graded,

x y u u u 1 k l Figure 7. A pearly trajectory contributing to the diﬀerential dx of the pearl complex.

LAGRANGIAN COBORDISM: RIGIDITY AND FLEXIBILITY ASPECTS 57 the degree corresponding to the critical points of f being given by their Morse index. The homology H∗(C(D),d) is independent of D (up to canonical isomorphisms) and is denoted by QH∗(L). Obviously this homology is also Z-graded. A monotone ∼ Lagrangian L is called narrow if QH(L) = 0 and it is called wide if QH(L) = H(L; K) ⊗ Λsee[7]. It is possible to define a version of Floer homology HF(L, L; Λ) with coefficients in Λ and there is an isomorphism, essentially due to Piunikin-Salamon-Schwartz [28], ∼ PSS : QH∗(L) = HF∗(L, L;Λ) . ∼ Further, in case L is exact, then QH∗(L) = H∗(L; Z2). Thus QH(L)isjustavariantofFloerhomology.Atthesametime,thisvariant is well-adapted to studying “individual” cobordisms. Indeed, let V :(L1,...,Li) ; → R (L1,...,Lk) be a cobordism. Consider a Morse function f : V so that the function is linear along the ends of V . Assume, for instance, that the negative gradient of f (with respect to some metric on V ) points “in” along the positive ends and points “out” along the negative ends. This is the typical picture of a function on a cobordism and the resulting Morse complex computes the singular ∪ ∪ Z homology H(V ; L1 ... Lk; 2). It is shown in [8] that by choosing an appropriate almost complex structure on C × M one can define a pearl complex, again over Λ, associated to this Morse function f. The resulting quantum homology is denoted ∪ ∪ by QH(V ; L1 ... Lk). Certainly, one can define similarly also the quantum homology QH(V ; L1 ∪ ...∪ Li)aswellas,bytakingf so that its negative gradient points “in” along all the ends of V , QH(V ), and, if f points “out” along all the ends, QH(V,∂V ). All these quantum homologies verify the expected dualities and other properties, just like their Morse counterparts, but more has to be true. Indeed, by Theorem 4.1 and its corollaries we know that the Floer homologies of the ends of a cobordism are related by a series of exact sequences. Given that Floer homology is related - via the PSS morphism - to quantum homology, the quantum homologies of the ends have to satisfy some stronger constraints compared to the respective Morse homologies. This is indeed the case and a prototypical example of this sort is next. Theorem 4.6. [8] Let L, L,L ∈L∗(M). i. If V : L ; L is a cobordism with V ∈L∗(C × M),thenQH(V,L)= 0=QH(V,L) and moreover QH(L) and QH(L) are isomorphic (via an isomorphism that depends on [V ]) as rings. If additionally L and L are wide, then the singular homology inclusions H1(L; Z2) → H1(V ; Z2) and H1(L ; Z2) → H1(V ; Z2) have the same image. When dim(L)=2,both ∼ these inclusions are injective and thus H1(L; Z2) = H1(L ; Z2). ii. Assume that W : L ; (L,L) is a cobordism with W ∈L∗(M).If QH(L) is a field (in other words, each element in QH(L) admits an inverse with respect to the quantum multiplication), then the inclusion QH(L) → QH(V ) is injective. Moreover, for each k we have the inequality:

rk(QHk(L)) ≤|rk(QHk(L1)) − rk(QHk(L2))| . ∗ Remark 4.7. For this result the condition on π1 in the deﬁnition of L is not actually necessary.

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An interesting particular case is when all Lagrangians and the cobordisms relating them are exact. In that case all the quantum homologies coincide with the respective singular homologies so that, for instance, the first point means that if V is an exact cobordism with a single exact positive end L and a single exact negative end L ,thenL → V and L → V are homology equivalences over Z2.In case this homology equivalence could be extended over Z and assuming in addition that L, L,V are simply connected and n ≥ 5, we deduce from the h-cobordism theorem that V is diffeomorphic to a trivial cobordism. A Lagrangian cobordism that is diffeomorphic to a cylinder is called a Lagrangian pseudo-isotopy. All of this sugests the following conjecture: an exact Lagrangian cobordism with one positive end that is exact and one negative end, also exact, is a pseudo-isotopy. An important step in this direction has been made recently by Suarez [33]: she shows that an exact Lagrangian cobordism as before that is also spin and so that the maps π1(L) → π1(V ), π1(L ) → π1(V ) are isomorphisms is indeed a pseudo-isotopy. Besides adjusting the arguments in Theorem 4.6 i so as to take into account orientations, her proof makes use of the Floer-theoretic Whitehead torsion introduced in [34] and of the s-cobordism theorem. An even stronger conjecture seems believable (but is, for the moment, in- tractable): a Lagrangian cobordism V : L ; L with V,L,L exact, is horizontally isotopic to a Lagrangian suspension.

5. Sketch of the construction of F We divide the presentation in two subsections: in the ﬁrst we explain the basic principles that are behind the machinery involved here; in the second subsection we list the main steps of the proof of Theorem 4.1.

5.1. Ingredients in elementary form. 5.1.1. Compactness and the open mapping theorem. The first indication that rigidity can be expected to play a significant role in the study of Lagrangian cobordisms - under the assumption of monotonicity - appeared in a paper of Chekanov ; [11]. His result is the following: assume that V :(L1,...,Lk) (L1,...,Ls)isa montone cobordism so that V is connected. Then all the Li’s and Lj ’s are monotone with the same monotonicity constant ρ, and moreover, they all have the same invariant dL (see §2.2). The monotonicity part of the claim is easy because the two morphisms: ω, μ : π2(M,Li) → R, Z are both seen to factor via ω, μ : π2(C × M,V ) → R, Z.The equality of the dL’s is much more interesting. For instance, it implies that if two monotone Lagrangians L, L with the same monotonicity constant have dL = dL , then they can not intersect in a single point. Indeed, by the surgery results from §2.3 two such Lagrangians are the end of a cobordism obtained as the “trace” of the surgery in the single intersection point. Here is the argument for the equality of the dL’s. First, fix some almost complex structure J on C × M so that, outside a set K × M where K ⊂ C is compact, the projection π : C × M → C is J− i holomorphic. We take K large enought so that π(V ) equals a union of horizontal lines outside of K as in Figure 8. Recall that dL counts the number ∈ Z2 of J-holomorphic disks of Maslov 2 through any (generic) point of L, in particular this number is independent of the point in L chosen to estimate it. We apply this remark to V and J. Pick one point P that belongs to

LAGRANGIAN COBORDISM: RIGIDITY AND FLEXIBILITY ASPECTS 59

Figure 8. The projection π : C × M → M is J− i holomorphic outside of K.

an end of V : P ∈ [a, +∞) ×{i}×Li ⊂ V ,andissothatp = π(P ) ∈ K (see §2.1). Consider a J-holomorphic disk with boundary on V , u :(D2,S1) → (C × M,V ) with P ∈ u(S1). Put v = π ◦ u. There is an open set U ⊂ D2 whose image by v avoids K.Letv = v|U : U → C\K.Inparticular,v is holomorphic. As it goes through p ∈ K and π(V ) is a union of horizontal lines outside of K,itiseasyto see that, by the open mapping theorem, v is constant. But this implies that v is constant and thus u has values in the ﬁber over p.Thus,u is actually a map 2 1 u :(D ,S ) → (M,Li). Assuming that the restriction of J to the ﬁbre over p is regular (which is easy to arrange) the conclusion is that dLi = dV .

Refinements of this argument are crucial in all the results discussed in this paper. The basic idea is to use again specific almost complex structures as J before so as to restrict the admissible behaviour of the J-holomorphic curves that are used in the definition of the Floer differential as well as in the other μk’s. This serves two purposes: it establishes compactness for the respective moduli spaces and, secondly, gives a particular form to the algebraic structures in question. As an example consider again Figure 5. Here is briefly how the definition of φV : HF(L, L) → HF(L, L) follows from these types of arguments. First we pick J so that π is J − i holomorphic outside of a compact set K very close to the “bulb” of V in the picture. In particular, the intersection points of γ × L with V are outside of K × M. We then define the Floer complex CF(γ × L, V ). The only issue with this definition is to make sure that the moduli spaces of J-holomorphic strips is compact. But our choice of J together with a simple application of the open mapping theorem, as before, implies easily this compactness. As a vector space CF(γ × L, V ) is isomorphic to CF(L, L) ⊕ CF(L, L). The differential in this complex is therefore a matrix: " # d φ (16) D = 1 . ψd2 If u is a holomorphic Floer strip contributing to D we let v = π ◦ u and notice that v is holomorphic outside K. In particular, it is holomorphic around the points where γ intersects π(V ). In view of this, using the open mapping theorem again as well as easy orientation arguments it is easy to deduce that d1 is the differential in 2 CF(L, L), d2 is the differential in CF(L, L )andψ = 0. Therefore, D = 0 implies that φ is a chain morphism and we put φV = H(φ).

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5.1.2. Using Hamiltonian deformations lifted from C. The second basic principle behind many of our proofs is that the algebraic structures deﬁned here - in particular HF(−, −) - are invariant with respect to horizontal isotopy and that by using various horizontal isotopies lifted from C one can get a variety of interesting relations. To exemplify how this principle is applied in practice we focus again on a situation similar to that in Figure 5 but this time in a simpler situation, when k = 1. In other words, we have a cobordism V : L ; L, V ∈L∗(M)andwe would like to notice that in this case the morphism φV : HF(L, L) → HF(L, L )is in fact an isomorphism (this is, of course, a very particular case of Theorem 4.1). For this purpose consider a second curve γ as in Figure 9. It is clear that γ and

Figure 9. γ × L and γ × L are horizontally isotopic.

γ are horizontally isotopic in the plane. Therefore, γ × L is horizontally isotopic to ∼ γ × L. We deduce HF(γ × L, V ) = HF(γ × L, V ) = 0 because γ × L ∩ V = ∅. But this means that the component φ of D in (16) is a quasi-isomorphism. 5.2. Outline of the proof of Theorem 4.1. 5.2.1. The Fukaya category of cobordisms. The fundamental step, and the one of highest technical difficulty, is to define a Fukaya category of cobordisms in R2 × F ∗ R2 × M which we denote ukcob( M). The objects in this category are therefore cobordisms V ∈L∗(C × M) and the morphisms Floer chains CF(V,V ). The construction follows the machinery in Seidel’s book [32] that is truly fundamental here. In particular, to deal with cobordisms that are non-transversal we use moduli spaces of curves verifying Cauchy-Riemann equations perturbed by Hamiltonian terms. One difference with the construction in [32] is that we work in a monotone setting and not an exact one. However, by arguments such as in, for instance, [7], the resulting issues are easily disposed off. A much more serious difficulty has to do with the compactness of the relevant moduli spaces, basically in continuation of the discussion in §5.1.1. The key issue is seen by looking to the presumtive morphisms from a cobordism V to itself. Thus we are considering the Floer chains CF(V,V ). Clearly, to be able to define such chains we need to use Hamiltonian perturbations that are non-compact. But this means that the curves u in our moduli spaces do not have the property that v = π ◦ u is holomorphic away from a compact set. Indeed, these v’s satisfy themselves some perturbed Cuachy-Riemann equations and the open mapping theorem does not apply to them directly. There are probably a variety of solutions to this issue but the one found in [9] is to pick very carefully the Hamiltonian perturbations so that the curves v can still be transformed by a change of variable - away from a large compact set - to holomorphic curves to which the open mapping theorem again applies.

LAGRANGIAN COBORDISM: RIGIDITY AND FLEXIBILITY ASPECTS 61

F F ∗ R2 × 5.2.2. Inclusion, triangles and . Once the category ukcob( M)isdeﬁned the proof proceeds as follows. Let γ : R → R2 be a curve in the plane with horizontal ends. There is an induced functor of A∞-categories: I F ∗ →F ∗ R2 × γ : uk (M) ukcob( M) deﬁned on objects by Iγ (L)=γ × L.

Fix a cobordism V : L ; (L1,...,Lk) as in Figure 10. Let MV be the Yoneda F ∗ R2 × module associated to V as in (7) but for the category ukcob( M). By using I Mγ F ∗ the functor γ we can pull back this module to a module V over uk (M), Mγ I∗ M V = γ ( V ). At the derived level, this module only depends on the horizontal 2 isotopy classes of V and γ. We consider a particular set of curves α1,...,αk ⊂ R M Mαi basically as in Figure 10. Therefore, we get a sequence of modules V,i := V , i =1,...,k.

Figure 10. A cobordism V together with curves of the type αi’s.

We then show that these modules are related by exact triangles (in the sense of triangulated A∞ categories): −1M →M →M →M ∀ ≤ ≤ (17) T Ls V,s−1 V,s Ls 2 s k. and that, moreover, there is a quasi-isomorphism φV : ML →MV,k.Thispointis certainly the heart of the proof and we will not attempt to explain it here besides indicating that, in essence, the exact triangles are deduced from arguments that eliminate certain behaviour of J-holomorphic polygons, somewhat similarly to how we noticed that the application ψ from (16) vanishes. Once these exact triangles are established, the definition of F is relatively direct, by translating the preceeding structures to the derived setting. Remark 5.1. It is an open question at this time how much the results described here - in particular, the construction of the functor F and the morphism Θ from (1) - can be extended beyond the montone case. Certainly, there are major technical difficulties with such an extension but this is not only a technical issue. Indeed, Theorem 4.1 implies that, for instance, if V is a monotone cobordism V : L ; L, ∼ then L and L verify HF(L, L) = HF(L ,L). Assuming that a reasonable notion of Floer homology HF(−, −) is defined in full generality the same argument would apply even if V is not monotone. But, as seen in our “flexibility” subsection §2.3, constructing general cobordisms V is easy without requiring monotonicity. As a consequence significantly different Lagrangians L and L would have the same

62 OCTAV CORNEA

HF(−, −). In short, we are here in front of an example of precisely the tension rigidity-ﬂexibility that was mentioned at the beginning of the paper: any invariant of type HF that is deﬁned in great generality can be expected to be quite weak.

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Department of Mathematics and Statistics, University of Montreal, C.P. 6128 Succ. Centre-Ville Montreal, QC H3C 3J7, Canada E-mail address: [email protected]

Contemporary Mathematics Volume 656, 2016 http://dx.doi.org/10.1090/conm/656/13076

Biochemical reaction networks: An invitation for algebraic geometers

Alicia Dickenstein

Abstract. This article is a survey of the recent use of some techniques from computational algebraic geometry to address mathematical challenges in systems biology. (Bio)chemical reaction networks deﬁne systems of ordinary differential equations with many parameters, which are needed for numerical simulations but that can be practically or provably impossible to identify. Under the standard modeling of mass-action kinetics, these equations depend poly- nomially on the concentrations of the chemical species. The algebraic theory of chemical reaction systems provides new tools to understand the dynamical behavior of (families of) chemical reaction systems by taking advantage of the inherent algebraic structure in the (parametric) kinetic equations.

1. Introduction Chemical Reaction Network Theory (CNRT) has been developed over the last 40 years, initially through the work of Horn and Jackson and subsequently by Feinberg and his students and collaborators [24–30, 50–53]. CRNT connected qualitative properties of ordinary differential equations corresponding to a reaction network to the network structure. In particular, assertions which are independent of specific parameter values have been obtained, in general assuming that all kinetics are of the mass-action form. New concepts were introduced, such as the deficiency of a reaction network, and several conditions were given on such networks for the existence, uniqueness, multiplicity and stability of fixed points. Fundational work has also been done by Vol’pert [79], with contributions with algebraic tools by Bykov, Kytmanov, Lazman and Yablonsky (see [8] and the references therein, together with more recent work as [44]). The principal current application of these developments is in the realm of biochemical reaction networks, that is, chemical reaction networks in biochemistry. Systems biology’s main goal is to understand the design principles of living systems. According to [36], the state of systems biology (at that moment, but still current) is like planetary astronomy science before Kepler and Newton and cannot be studied without mathematics and physics.

2010 Mathematics Subject Classiﬁcation. Primary 14Q99, 13P15, 92C42, 92C45. AD was partially supported by UBACYT 20020100100242, CONICET PIP 20110100580, and ANPCyT PICT 2013-1110, Argentina.

c 2016 American Mathematical Society 65

66 ALICIA DICKENSTEIN

Recent work has focused on long-term dynamics as well as the capacity for multiple equilibria and how such dynamics depend on the specific rate parameters, mainly manipulating R-linear combinations of the polynomials defining the dynamical systems (or equivalently, studying the kernel of the matrix M in (6)) [13, 15, 17, 18, 21, 42, 62, 72–74, 76]. We can use algebraic geometry to analyze systems biology models. Symbolic treatment of the parameters does not need a priori determination (which can be practically and theoretically impossible [17,54]), as numerical simulations require. Karin Gatermann introduced the connection between mass-action kinetics and toric varieties at the beginning of the last decade [38–40]. Gunawardena also started approaching results from CRNT with algebraic tools over the last years [45–47,61, 77, 78]. In joint work with Craciun, Shiu and Sturmfels, we studied in [12] toric dynamical systems (aka complex balanced mass-action systems) with an algebro- geometric perspective. The steady state locus of these systems coincides with the real points of a toric variety and, in appropriate coordinates in parameter space, the equations describing these complex balanced systems are also binomial. Advanced algebraic tools have been introduced by different authors over the last years [16, 22, 23, 31, 32, 34, 43, 48, 58, 60, 66, 69, 75]. Almost all cells in a body have the same genetic information. Multistationarity (see Definition 2.4) provides a mechanism for switching between different response states in cell signaling systems and enables multiple outcomes for cellular-decision making [59]. Questions about steady states in biochemical reaction networks under mass-action kinetics are fundamentally questions about (nonnegative) real solutions to parametrized polynomial ideals. We present in Section 2 the basic notations and concepts about chemical reaction networks. Section 3 concentrates on the important enzymatic networks, that we use to exemplify questions on multistationarity. Section 4 is devoted to the notion of steady state invariants. Invariants depending on selected variables can be used to understand the design of the different mechanisms. We distinguish four levels of invariants and we show applications to model selection, to study absolute concentration robustness and to obtain nontrivial bounds via implicit dose-response curves. It follows that the study of ideals over polynomial rings unveils features of the steady states not visible working only with coefficients in R, but further tools from real algebraic geometry are required. Finally, in Section 5 we summarize recent general results on sign conditions for multistationarity, that hold beyond the framework of chemical reaction networks. Along the text, we recall results from my joint recent papers and preprints [58, 65, 67–69]. A more comprehensive account will appear in the book in progress with Elisenda Feliu [19]. We end this introduction with pointers to a few important subjects we have not addressed in this text, together with an overview of general goals for our approach and new algebro-geometric tools that we expect to incorporate. All biological processes are complex and involve many variables and (unknown) reaction rate constants. An apparent solution to the complexity challenges in cellular networks consists of studying smaller subunits that one can analyze separately. In fact, essential qualitative features of biological processes can usually be understood or qualitatively approximated for parameters in a certain range, in terms of a small number of crucial variables [59]. In [71], the authors defined network motifs as patterns of interconnections that recur in many different parts of a big

BIOCHEMICAL REACTION NETWORKS 67 network. Study of subnetworks to determine multistationarity has been addressed for instance in [9](viaelementary flux modes)and[35, 57](viaversionsofthe implicit function theorem). We expect that tools from deformation theory could help extending these results to the case of degenerate steady states. Differential algebra methods and in particular differential elimination methods, provide tools for searching hidden relations which are consequences of our differential-algebraic polynomial (nonlinear) equations. They have been used for parameter estimation in nonlinear dynamical systems and model reduction of biochemical systems (via implicit quasi-steady state approximation) and some related software is available [3–7]. It would be interesting to further explore the use of these tools. We have not discussed the global dynamic behaviour of the systems. The main open conjecture in the field of Chemical Reaction Network Theory is the Global Attractor Conjecture, which dates back to the early 1970s. Complex balanced chemical reaction networks associated with weakly reversible graphs, possess a unique positive steady state in any given stoichiometric compabitility class (see Section 2), which shows local asymptotic stability deduced from the existence of a Lyapunov function. The Global Attractor Conjecture asserts that this is in fact a global attractor for the dynamics. This statement is proven in the absence of steady states with zero coordinates, in case the reaction graph is connected or in case the dynamics occurs in dimension at most three, but the combinatorics of zero coordinates of the boundary steady states makes the search for a proof of the general result highly complicated [1,12,16,55]. At the time of the revision of this article, G. Craciun has posted a first version of an article which would positively solve the Conjecture [11]. Tools from elimination theory in computational algebraic geometry and from real algebraic geometry can be used to study the number and stability of steady states in families, as well as the possible occurrence of bifurcations and oscillations in polynomial (nonlinear) dynamical systems. One general goal is to partition the positive orthant in constant rate space for a given biochemical network into semialgebraic sets, in such a way that on each chamber the dynamic behaviour can be determined. The study of properties that depend on the structure of the network and are independent of the particular reaction rate constants in this semialgebraic decomposition of parameter space, would allow to see “the woods” and not “only the trees”. The super goal is to understand the basic mechanisms in nature for multistationarity and for oscillations. In theory, computational algebraic geometry can give many answers. In practice, these responses tend to be too complex to be understood or computed. Many answers are missing and require the combination of tools from computer algebra, real algebraic geometry, numerical algebraic geometry, discrete mathematics, dynamical systems, and biochemistry!

2. Basics on chemical reaction networks (CRN) We start with a simple but meaningful example of a biochemical reaction network: the T-cell signal transduction model proposed by the immunologist McKei- than [63]. The main task of the immune system is to recognize that a strange body has entered the organism. T-cell receptors bind to both self-antigens and foreign antigens and the dynamical features of this model give a possible explanation of how T-cells can be sensitive and speciﬁc in recognizing self versus foreign antigens.

68 ALICIA DICKENSTEIN

A mathematical study of the dynamics of this network was done by Sontag in [76]. In its simplest case, the network of reactions is as follows:

A +KKeKBK rr8KKKK κ rr KKκKK21 31rr KKKK rrr κ12 KKKK or % DC, κ23 where A denotes the T-cell receptor protein, B denotes the Major Histocompatibil- ity protein Complex (MHC) of antigen-presenting cell, C denotes the biochemical species A bound to species B,and D denotes an activated (phosphorylated) form of C. The binding of A and B forms C, which undergoes a modification into its activated form D before “transmitting a signal” (that is, before participating in another chemical reaction). The general mechanism proposed by McKeithan includes several activated forms of C, until a final active form that “triggers the attack” to the foreign antigen is obtained. This biochemical reaction network has: • r =4reactions among • m =3complexes A + B, C,and D, which are composed by • s =4species A, B, C, D,and • r =4reaction rate constants κ12,κ21,κ23,κ31 ∈ R>0 attached to the different reactions. A kinetics is then attached to this labeled directed graph to describe how the concentrations xA,xB,xC ,xD of the different biochemical species evolve in time. McKeithan assumes that the vector of concentrations x(t)= (xA(t),xB(t),xC (t),xD(t)) evolves according to mass-action kinetics, which is a modeling commonly used in chemistry and biology when there are sufficiently many molecules that are well mixed. The Law of Mass Action was proposed by two Nor- wegians: Cato Guldberg (1836–1902), a chemist, and Peter Waage (1833–1900), a mathematician, in an article published in Norwegian in 1864. Their work was then published in French in 1867 and finally, a fuller and further developed account appeared in German in 1879, and was then recognized (in the meantime this principle was rediscovered by van’t Hoff). The Law of Mass Action is derived from the idea that the the rate of an elementary reaction is proportional to the probability of collision of reactants (under an independence assumption), that is, to the product of their concentrations. We write the precise formulation in (1) below. The explicit differential equations for the concentrations x(t) in the T-cell signal transduction model are the following:

dxA − − (1,1,0,0) (0,0,1,0) (0,0,0,1) dt = κ12xAxB + κ21xC + κ31xD = κ12x + κ21x + κ31x dxB − − (1,1,0,0) (0,0,1,0) (0,0,0,1) dt = κ12xAxB + κ21xC + κ31xD = κ12x + κ21x + κ31x dxC − − (1,1,0,0) − (0,0,1,0) dt = κ12xAxB κ21xC κ23xC = κ12x (κ21 + κ23)x dxD − (0,0,1,0) − (0,0,0,1) dt = κ23xC κ31xD = κ23x κ31x . In general, the starting data for a chemical reaction network are a ﬁnite set of s species (whose concentrations x1,...,xs will be our variables), a ﬁnite set of r κij reactions (labeled edges i → j,whereκij ∈ R>0 are the reaction rate constants), ∈ Zs between m complexes y1,...,ym ≥0 among the species (which are classically represented as nonnegative integer combinations of the species and which give rise

BIOCHEMICAL REACTION NETWORKS 69

y y yi i1 i2 ··· yis to monomials in the concentrations of the chemical species x = x1 x2 xs ). The entries of the complexes are called stoichiometric coeﬃcients. Definition 2.1. A chemical reaction network (CRN) is a ﬁnite directed graph

G =(V,E,(κij)(i,j)∈E, (yi)i=1,...,m) whose vertices are labeled by complexes and whose edges are labeled by positive real numbers. Mass-action kinetics specified by the network G gives the following autonomous system of ordinary differential equations in the concentrations x1,x2,...,xs of the species as functions of time t: dx (1) = κ xyi (y − y ). dt i,j j i (i,j)∈E Note that system (1) is of the form dx (2) k = f (x),k=1,...,s, dt k where f1,...,fs are polynomials in R[x1,...,xs]. A first natural question is which autonomous polynomial dynamical systems come from a CRN under mass-action kinetics. The answer is due to Hárs and Tóth: Lemma 2.2 ([49]). A polynomial dynamical system dx/dt = f(x) in s variables (x1,...,xs) arises from a CRN under mass-action kinetics if and only if there exists real polynomials pk,qk,k =1,...,s,withnon negative coefficients such that fk = pk − xkqk for all k. The necessary condition that each monomial with negative coefficient in the polyonomial fk has to be divisible by xk is straightforward from (1). The converse is constructive. One interesting feature that follows from this constructive proof is the fact that the polynomials fk do not determine the network, only (almost) the source complexes of the reactions (those labeling the initial node of a directed edge). We refer the reader to [17, 54] for extensions and precisions of Lemma 2.2, in particular, identifiability of the reaction rate constants κij for a given network. In general, one assumes the structure of the reaction network and would like to infer dynamical properties of the system from this structure, even if most reaction rate constants are unknown. The restriction on the coefficients of a CRN under mass-action kinetics given by Lemma 2.2 is satisfied for instance by the oscillatory Lotka-Volterra equations, but not by the “chaotic” Lorenz equations dx dx dz 1 = αx − αx , 2 = γx − x − x x , = x x − βx , α,β,γ ∈ R , dt 2 1 dt 1 2 1 3 dt 1 2 3 >0 due to the existence of the term −x1x3 in f2. Definition 2.3. The steady state variety V (f) of the kinetic system (2) equals the nonnegative real zeros of f1,...,fs, that is, the nonnegative points of the real algebraic variety cut out by f1,...,fs. Any element of V (f)iscalledasteadystate of the system.

Note that the positive solutions of the system x1f1 = ··· = xsfs =0equal the positive solutions of f1 = ···= fs = 0 (but of course the dynamics of the corresponding diﬀerential systems is diﬀerent). So, any system of s real polynomials

70 ALICIA DICKENSTEIN in s variables deﬁnes the positive steady states of a CRN under mass-action kinetics. However, realistic models have particular features that allow for interesting particular results. We will focus in particular on enzymatic networks. Another direct consequence of the form of the equations in (1) is that for any dx trajectory x(t), the vector dt lies for all t in the so called stoichiometric subspace S, which is the linear subspace generated by the diﬀerences {yj −yi | (i, j) ∈ E}.Using the shape of the polynomials fk = pk − xkqk in Lemma 2.2, it is straightforward to see that a trajectory x(t) starting at a nonnegative point x(0) lies in the closed ∩Rs ≥ polyhedron (x(0)+S) ≥0 for all t 0, called a stoichiometric compatibility class. The (linear) equations of x(0) + S are called conservation relations.

Different stoichiometric compatibility classes Note that for any autonomous dynamical system of the form (2), any linear s relation i=1 cifi = 0 with real coefficients c1,...cs, gives rise to the restriction s that i=1 cixi has to be constant along trajectories. In our setting, the linear equations for S give conservation relations, but for specific f1,...,fs there could be further linear constraints. As we pointed out in the introduction, a central notion is the following: Definition 2.4. We say that system (1) exhibits multistationarity if there exist at least two steady states in the same stoichiometric compatibility class. The following figure illustrates the intersection of the steady state variety V (f) with different stoichiometric compatibility classes. The middle one has 3 different steady states x(1),∗,x(2),∗,x(3),∗, so the system exhibits multistationarity.

In the following section we will concentrate on multistationarity questions of enzymatic networks. Section 5 presents recent general mathematical results to preclude or allow the occurrence of multistationarity based on sign vectors.

3. Enzymatic networks The Nobel Prize in Physiology or Medicine 1992 was awarded jointly to Edmond H. Fischer and Edwin G. Krebs “for their discoveries concerning reversible protein

BIOCHEMICAL REACTION NETWORKS 71 phosphorylation as a biological regulatory mechanism”. Phosphorylation/dephosphorylation are post-translational modification of proteins mediated by enzymes, particular proteins that add or take off a phosphate group at a specific site, inducing a conformational change that allows/prevents the protein to perform its function. The standard building block in cell signaling is the following enzyme mechanism, which is called a Michaelis-Menten mechanism, named after the German biochemist Leonor Michaelis and the Canadian physician Maud Menten. This basic network involves four species: the substrate S0, the phosphorylated substrate S1, the enzyme E and the intermediate species ES0. The enzyme E is not “consumed” after the whole mechanism, which is assumed to be with mass-action kinetics. The concentration of the donor of the phosphate group is considered to be constant, thus hidden in the reaction rate constants and ignored. A scheme is as follows, with the 3 reaction rate constants called kon,koff ,kcat:

kon −→ kcat (3) S0 + E ←− ES0 → S1 + E koﬀ

S 0 S1

ES ES E 0 0 E

One canonical class of biological systems exhibiting multistationarity are protein kinase mechanisms that involve multiple phosphorylation of a substrate. There are (substrate) proteins in humans that are known to have more than 150 possible phosphorylation sites [78]. The following CRN corresponds to the case of n =2sequential phosphorylations:

kon0 kon1 −→ kcat0 −→ kcat1 (4) S0 + E ←− ES0 → S1 + E ←− ES1 → S2 + E koﬀ0 koﬀ1

lon1 lon0 −→ lcat1 −→ lcat0 S2 + F ←− FS2 → S1 + F ←− FS1 → S0 + F loﬀ1 loﬀ0

This network involves nine species: the substrates with zero, one and two phosphorylated sites S0,S1,S2 (known as phosphoforms), the intermediate species ES0, ES1,FS1,FS2 plus two enzymes E,F (E is called a kinase and F a phosphatase), and ten complexes denoted as integer linear combinations of species by S0 +E,S1 + E,S2 + E,ES0,ES1,S0 + F, S1 + F, S2 + F, FS1,FS2. Renaming the variables and

72 ALICIA DICKENSTEIN the complexes following the previous ordering, we get the following dynamical system for the concentrations under mass-action kinetics: dx dx 1 =−k x x + k x + l x 6 =l x x − (l + l )x dt on0 1 8 off0 4 cat0 6 dt on0 2 9 cat0 off0 6 dx dx 2 =−k x x + k x + k x 7 =l x x − (l + l )x dt on1 2 8 cat0 4 off1 5 dt on1 3 9 cat1 off1 7 dx −l x x + l x + l x 8 =−k x x − k x x +(k + k )x on0 2 9 off0 6 cat1 7 dt on0 1 8 on1 2 8 off0 cat0 4 dx 3 =k x − l x x + l x +(k + k )x dt cat1 5 on1 3 9 off1 7 off1 cat1 5 dx dx 4 =k x x − (k + k )x 9 =−l x x − l x x +(l + l )x dt on0 1 8 off0 cat0 4 dt on0 2 9 on1 3 9 cat0 off0 6 dx 5 =k x x − (k + k )x +(l + l )x dt on1 2 8 off1 cat1 5 cat1 off1 7

The stoichiometric subspace S has codimension 3, so there are 3 linearly independent conservation relations, usually taken as total substrate, total kinase and total phosphatase:

x1 + x2 + x3 + x4 + x5 + x6 + x7 =Stot

x4 + x5 + x8 =Etot

x6 + x7 + x9 =Ftot. So, there are only 6 linearly independent diﬀerential equations in the system. The constants (Stot,Etot,Ftot) are determined by the initial conditions. We see that each stoichiometric compatibility class is compact since adding the 3 conservation relations we get a positive linear combination involving all the variables equal to a positive number, so each class is bounded (and closed). In general, the n-site phosphorylation system is of great biochemical importance: it is a recurring network motif in many networks describing biochemical processes. The common zeros of f1,...,fs equal the common zeros of the ideal of their polynomial consequences (the steady state ideal):

(5) If = {g1f1 + ···+ gsfs : gi ∈ R[x1,...,xs],i=1,...,s}.

The polynomials f1,...,fs are generators of If . We refer the reader to [10]forthe basic notions of polynomial ideals and Gr¨obner bases. If the steady state ideal If is a binomial ideal, that is, if it can be generated by polynomials which are binomials (i.e., polynomials with two terms), we say that the system has toric steady states. We prove in [69] that the chemical reaction system associated with the multisite n-phosphorylation of a protein by a kinase/phosphatase pair in a sequential and distributive mechanism (with the same structure as the mechanism in (4) but with n sites) has toric steady states.This result was implicit in [80] and it is a particular case of [77]. This system has 3n +3 species, 4n + 2 complexes and still 3 linearly independent conservation relations. Wang and Sontag studied in [80] the number of steady states in the general n-site sequential distributive phosphorylation network and showed that there are at most 2n − 1 steady states in each stoichiometric compatibility class. They also identiﬁed a particular open set in the positive orthant of the constant rate space Rr >0 where the number of positive steady states in the same compatibility class is

BIOCHEMICAL REACTION NETWORKS 73 n +1 (for n even) or n (for n odd) and conjectured that the maximum possible number is n +1 forany n. Very recently, it was shown in [37], that in fact for any n = 3 and 4 there can be up to 2n − 1 stoichiometrically compatible positive steady states, for particular choices of the reaction rate constants. Even for n =3 it is very complicated to give a precise description of the (semialgebraic) regions in Rr which >0 can be partitioned according to the maximal number of steady states and no study explains for the moment how many of the known steady states in a given compatibility class are attractors for the dynamics. We show in [68] that many other important networks have toric steady states including most of the motifs of enzyme cascades studied in [33], for example, the following cascade of phosphorylations known as the MAPK/ERK pathway: E

S0 S1

F1 P0 P1 P2

F2 F2

R0 R1 R2

F3 F3 Each curved arrow in this diagram represents a digraph with 3 nodes as in (3), where the enzyme is the label of the arrow. Note that the phosphorylated (or double phosporylated) substrate in the upper reactions acts as an enzyme down the cascade. In general, deciding multistationarity amounts to the difficult question of determining emptiness of a (complicated) semialgebraic set, which is in principle algorithmic but unfeasible in practice. For chemical reaction systems with toric steady states for all choice of positive reaction rate constants, we have the following explicit criterion for multistationarity [69]. First, if the system has toric steady states for all choice of positive reaction rate constants, the steady states can be explicitly parametrized by monomials (or shown to be empty). That is, we can check for non emptiness and the positive steady states can be parametrized by a monomial map v1 vs t → (c1(κ)t ,...,cs(κ)t ), ∈ Rd where t +, d is the dimension of the steady state variety and c1,...,cs are rational functions of the κij. Now, we can check for multistationarity in an algorithmic way (under the conditions detailed in [69, Section 3]). Call V ∈ Nd×s the matrix with columns v1,...,vs. The following is a simplified version of Theorem 5.5 in [69]: Theorem 3.1 ([69]). Fix a chemical reaction network G with s species, under mass-action kinetics such that there exist positive constants μij for all reactions 1 such that μij(yj − yi)=0. Assume the system has toric steady states for all reaction rate constants and it satisfies Condition 3.1 or 3.16 in [69].LetV ∈ Nd×s be a matrix giving the exponents of a parametrization of the positive steady state variety. There exists a reaction rate constant vector such that the resulting

1Note that by (1), this condition is necessary for the existence of a positive steady state

74 ALICIA DICKENSTEIN chemical reaction network exhibits two diﬀerent positive steady states in the same stoichiometric compatibility class if and only if there exists an orthant O of Rs of any positive dimension that the two intersections O ∩ image(V ) and O ∩ S are both non empty, or in other words, if and only if there exist non-zero α ∈ image(V ) and β in the stoichiometric subspace S with sign(αi)=sign(βi) for all i =1,...,s. We will present a recent general result on multistationarity in Section 5.

4. Steady state invariants We keep in this section the notations of Section 2. Note that we can also write the polynomial autonomous system (1) which models the kinetics of a chemical reaction network, as a real matrix M ∈ Rs×m multiplied by a vector of monomials Ψ(x) with i-th coordinate equal to xyi : dx (6) = f(x)=M(Ψ(x)). dt Definition 4.1. A steady state invariant (or simply, an invariant) is a polynomial that vanishes on the steady state variety V (f).

The given polynomials f1,...,fs are trivially steady state invariants. But we are interested in describing new invariants that reveal further properties of the system. In many cases, it is most important to find invariants that only depend on a selected subset of variables, which usually correspond to those concentrations that are easier to measure, or to concentrations one wants to relate at steady stated. We can distinguish four “levels” of invariants. Level 1: Any element of the rowspan of M defines an invariant which is an R- linear combination of f1,...,fs. Level 1 invariants depending on fewer complexes can be simply obtained by Gaussian elimination. For any ele- m ment λ in the rowspan of M,thesum i=1λivi vanishes for any vector m ∈ yi v ker(M); in particular, the polynomial i=1 λix vanishes at steady state. Level 2: Any polynomial in the steady state ideal If ⊂ R[x1,...,xs]definedin(5) is an invariant, which can be obtained via computational algebraic geometry methods as a polynomial linear combination of f1,...,fs.Inparticu- lar, any invariant of Level 1 is an invariant of Level 2, and the inclusion is strict. Note that any invariant of Level 2 vanishes on all complex common zeros of f1,...,fs. Elimination ideals If ∩ R[xi,i ∈ Γ] for a given subset Γof{1,...,s}, can be effectively computed with Gröbner basis methods, which are for instance efficiently implemented in the free computer algebra systems Singular [20] or Macaulay2 [41]. We will mainly deal with positive steady states, which in particular have nonzero coordinates. Primary decomposition of ideals has been applied in [75] to describe boundary steady states (with some zero coordinate). Level 3: Any polynomial in the radical If of the ideal If is an invariant. By Hilbert Nullstellensatz, these are precisely those polynomials that vanish on all the complex common zeros of f1,...,fs. The radical ideal If can also be computed via computational algebraic geometry, keeping the same zeros but without “multiplicity”. Level 4: Any polynomial which vanishes on V (f), that is on the nonnegative real zeros of If , is an invariant by definition. These polynomials form an ideal

BIOCHEMICAL REACTION NETWORKS 75 R ≥0 If that we could call the positive real radical of If .Thepositivereal R radical is in turn contained in the real radical If of If composed of all polynomials which vanish on the real zeros of If . These notions pertain to the (diﬃcult) realm of real algebraic geometry. In general, we have that R R≥ (7) If ⊂ If ⊂ If ⊂ 0 If , and the inclusions are in general strict. A simple example for n =1isgivenbythe ⊂ R 2 2 − 2 ideal If [x] generated by the polynomial f = x (x 1)(x + 1). In this case, 2 − 2 \ 2 − R \ x(x 1)(x + 1) lies in If If , x(x 1) lies in If If , the polynomial R R x(x − 1) lies in ≥0 If \ If and x − 1 vanishes on the positive real zeros of If . Another simple example in one variable shows that algebraic extensions enter 5 R into the picture: for instance, take f = x − 2x;then ≥0 If can be generated by the polynomial x2 − αx, with the additional information that α2 − 2=0and α>0. However, the containments in (7) are equalities for the most usual enzymatic networks. Invariants can be used to check the (un)correctness of a proposed model [61]. A baby example of this application taken from [46] is the following. In the sequential enzymatic mechanism for n = 2, we get by elimination of variables an invariant − 2 of Level 2 of the form xj (x1x3 Kx2) with j =8orj =9,whereK depends on the (unknown) reaction rate constants but not on the initial conditions! Recall that x1,x2,x3 denote the concentrations at steady state of the unphosphorylated, singly phosphorylated or doubly phosphorylated substrate and x8,x9 denote the enzymes, which can be measured and are assumed to be positive. So the “values at steady state” (x1,x2,x3) of the concentrations for diﬀerent runsnshould satisfy 2 for this model that the points (x1x3,x2) lie (approximately) on a line. Even if the slope is unknown, plotting these points allows to check the correctness of the model. In fact, K is the following explicit rational function in the reaction rate constants (obtained via elimination in the polynomial ring with variables xi and κij): it is the quotient P1/Q1 of the following polynomials:

P1 = κ10,7κ25κ41κ54κ79κ96 + κ10,7κ25κ42κ54κ79κ96 + κ10,8κ25κ41κ54κ79κ96

+κ10,8κ25κ42κ54κ79κ96,

Q1 = κ10,7κ14κ42κ52κ8,10κ96 + κ10,7κ14κ42κ52κ8,10κ97 + κ10,7κ14κ42κ53κ8,10κ96

+κ10,7κ14κ42κ53κ8,10κ97. Note that steady states correspond to nonnegative constant solutions of (1), so if for any finite t the system is at steady state, then the trajectory is constant. If s dxi there exists a positive conservation relation i=1 ci dt =0withc1,...,cs > 0as in this example, the trajectories are bounded (so the system is conservative)and then each trajectory is defined for any t ≥ 0. The “values at steady state” are the limit values limt→∞ xi(t) (when these limits exist), which can be approximated with experimental measurements. 4.1. Invariants and the notion of Absolute Concentration Robust- ness. Shinar and Feinberg introduced in [74] the notion of Absolute Concentration Robustness (ACR, for short) of a given chemical species xj . This happens when the j-th coordinate of the positive steady states of the system have a fixed value, independent of the given positive steady state and even independent of the value of the conservation relations (see also [72] for the notion of robustness of the output with respect the initial conditions). This is a very peculiar feature that shows up

76 ALICIA DICKENSTEIN in real examples. Here is a particular mechanism extracted from [74]. The enzyme X is a kinase (known as EnvZ) present in the the bacteria Escherichia Coli, that can be self-transformed into XD and XT and it can then be self-phosphorylated to produce the species Xp.InXp form, it can react with species Y (known as OmpR) to obtain the phosphorylated form Yp, while XD and XT can dephosphorylate Yp by the standard Michaelis-Menten mechanism:

κ12 κ23 κ34 XD X XT → Xp κ21 κ32 κ56 κ67 Xp + Y XpY → X + Yp κ65 (8) κ 89 κ9,10 XT + Yp XTYp → XT + Y κ98 κ11,12 κ12,13 XD + Yp XDYp → XD + Y κ12,11

We denote by x1,...,x9 the species concentrations as follows: xXD = x1,xX = x2,xXT = x3,xXp = x4 ,xY = x5,xXpY = x6,xYp = x7,xXTYp = x8,xXDYp = x9 . In fact, this system has toric steady states. Indeed, an ideal is binomial if and only if any reduced Gr¨obner basis is composed of binomials. Any such basis gives a binomial system of generators for the ideal. The reduced Gr¨obner basis of If with respect to the lexicographical order x1 >x2 >x4 >x5 >x6 >x8 >x9 >x3 >x7 consists of the following binomials:

g1 =[κ89κ12κ23κ9,10(κ12,11 + κ12,13)+κ11,12κ21κ12,13(κ98 + κ9,10)(κ32 + κ34)]x3x7− −[κ23κ34κ12(κ12,11 + κ12,13)(κ98 + κ9,10)]x3 g2 =[−κ11,12κ21κ34(κ98 + κ9,10)(κ32 + κ34)]x3+ +[κ11,12κ21κ12,13(κ98 + κ9,10)(κ32 + κ34)+κ12κ23κ89κ9,10(κ12,11 + κ12,13)]x9 g3 =[−κ23κ34κ89κ12(κ12,11 + κ12,13)]x3+ +[κ23κ9,10κ89κ12(κ12,11 + κ12,13)+κ11,12κ21κ12,13(κ98 + κ9,10)(κ32 + κ34)]x8 g4 = κ67x6 − κ34x3 g5 = κ56κ67x4x5 + κ34(−κ65 − κ67)x3 g6 = κ23x2 +(−κ32 − κ34)x3 g7 = −κ21(κ32 + κ34)x3 + κ12κ23x1

Note that g1 has the form g1 = Q1x3 −Q2x3x7 = x3(Q1 −Q2x7), where Q1 and Q2 are homogeneous polynomials in the reaction rate constants which are positive for positive values of the κij.Thus,any positive steady state satisﬁes that the value of x7 = xYp equals Q1/Q2, independently of the initial concentrations. So the Level 2 invariant g1 ∈ If shows immediately that the system exhibits ACR in Yp. Note that the two monomials that occur in g1 correspond to two complexes in our network, so one could imagine that it is possible to get a binomial only involving x3x7 and x7 as a Level 1 invariant, that is, via R-linear combinations of f1,...,f9 However, we prove in [69] that this is not possible and that Level 1 invariants cannot reveal the ACR property.

4.2. Invariants and robust bounds. Most of the literature on chemical reaction networks only deals with the computation of Level 1 steady state invariants, that we call Type 1 Complex Invariants in [58], from where we extracted the following examples of CRN with diﬀerent bifunctional enzymes. Being bifunctional means that the same enzyme has two diﬀerent binding sites in such a way that the enzyme can both catalyze a phosphorylation, or the reverse dephosphorylation.

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The ﬁrst example is a biologically plausible modiﬁcation of the network (8): we add a reaction with the self dephosphorylation Yp −→ Y of Yp. The resulting system does not have toric steady states. We show in [58] that there is no longer

ACR behaviour in the variable x7 = xYp , but instead we found a particular Level 1 invariant depending on a selected proper subset of the complexes, that allowed us to find two nontrivial bounds at steady state independent of initial conditions and the total amounts of the enzymes. According to the numbering of the nodes in [58], we have that for any positive steady state, {k1 k3k5 (k14 + k15) (k11 + k12)} xYp < min ,k5 . k2 (k4 + k5) k13k15 k10k12 The factors in these bounds as well as the particular reaction rate constants entering the expressions can be biochemically interpreted. The second example corresponds to a bifunctional enzyme (known as PFK2- F2,6BPase) in a mammalian cell. In this case, we again get in [58] from Level 1 particular invariants depending on some chosen complexes and the signs of their coefficients, a robust bound in the concentration of a smaller enzyme known as fructose-6-phosphate or in the concentration of the enzyme called F2,6BP, depending on the sign of the difference of two specific reaction rate constants, independently of the stoichiometric compatibility class [18, 58]. 4.3. Invariants and implicit dose–response curves. We say that b>0is a trivial upper bound for the ith species if there exists a conservation relation a1x1 + a2x2 +···+xi +···+asxs = b with all aj ≥ 0. Note that b is an upper bound for the concentration of xi all along the trajectory. Using invariants and elimination, we show in [67] how to improve these bounds for steady state concentrations of specific species of the system that are usually considered as the output, and so we bound what is called the maximal response of the system, regardless of the occurrence of multistationarity. For example, the concentration of the doubly phosphorylated substrate x3 = xS2 can be taken as the output in the sequential phosphorylation mechanism with two sites. The input of the system is in general a quantity that depends on the initial concentrations, for instance the total amount Etot of the kinase, that can be usually regulated. Denote by σ the codimension of the stoichiometric subspace S and suppose f1,...,fs−σ are linearly independent. Choose also σ independent conservation relations 1 − c1,...,σ − cσ (where 1,...,σ are homogeneous linear forms and ci are constants). Fix the values of c2,...,cσ and take c = c1 as our input and x1 as our output variable. We assume, as it is in general tacitly assumed, that there are a (nonzero) finite number of (complex) solutions to the equations

(9) f1 = f2 = ···= fσ = 1 − c = 2 − c2 = ···= σ − cσ =0, for any value of c. In particular, there are a (nonzero) ﬁnite number of points in the intersection of V (f) with each stoichiometric compatibility class. It can be seen that there exists a nonzero polynomial p = p(c, x1) in the ideal generated by the polynomials f1,f2,...,fs,1−c, 2−c2,...,σ −cσ in R[c, x1,...,xs], depending ony on x1 and c and with positive degree in x1, which can be computed with standard elimination tools in computational algebraic geometry (see Lemma 2.1 in [67]). The curve C = {p =0} gives the implicit relation between the input and output variables at steady state, that we call an implicit dose–response curve, extending the name of dose–response curve usually given in case x1 can be analytically expressed in terms

78 ALICIA DICKENSTEIN of c. In the general case, p has high degree both in c and in x1 andnosuchexpression is available. However, if one is able to plot the curve C = {p =0}, then an upper bound for the values of x1 at steady state can be read from this plotting, but an implicit plot has in general bad quality and is inaccurate. Instead, one can appeal to the properties of resultants and discriminants to preview a “box” containing the intersection of C with the ﬁrst orthant in the plane (x1,c). This gives improved bounds which yield smaller starting boxes to launch numerical computations. We moreover illustrate in the application to the enzymatic network studied in [62], the relation between the exact implicit dose-response curve we obtain symbolically and the standard hysteretic diagram provided by a numerical solver that is currently seen in the literature. The setting and tools we propose in [67] could yield many other results adapted to any autonomous polynomial dynamical system.

5. General results on sign conditions and multistationarity Uniqueness of positive solutions plays an important role in many applications and domains of mathematics, beyond chemical reaction networks. In the recent joint paper [65], we were able to isolate and generalize many previous results, in particular, Birch’s theorem [2] in Statistics and Feinberg’s theorem for complex balanced equilibria in case of deficiency zero [27, Prop. 5.3 and Cor. 5.4], as well as Theorem 3.1 above (together with several other results quoted in [65]). The setting is as follows, where n and m refer to any two natural numbers (so m does not denote in this section the number of complexes, and n could be s, d or any other suitable number of variables). Rn → Rm Consider a family of generalized polynomial maps fκ : >0 defined on m×r the positive orthant, associated with two fixed real matrices A =(aij) ∈ R , ∈ Rr×n ∈ Rr B =(bij) ,andr real positive parameters κ >0: r bj1 bjn (10) fκ,i(x)= aij κj x1 ...xn ,i=1,...,m. j=1 Note that we allow real exponents and not only nonnegative integer exponents. Definition . Rn → Rm 5.1 We say that fκ : >0 is injective with respect to ⊂ Rn ∈ Rn − ∈ a subset S if for all distinct x, y >0 such that x y S we have f(x) = f(y). Clearly, if we have a mass-action kinetics system (2), which is injective with respect to the stoichiometric susbpace S according to Definition 5.1 with fκ = f, then there cannot be two different positive steady states on any stoichiometric compatibility class. The following result is a simplified version of Theorem 1.4 in [65], where we also discuss the algorithmic issues. We need the following notations. The sign vector σ(x) ∈{−, 0, +}n of a vector x ∈ Rn is defined componentwise. Given a subset T , σ(T ) denotes the set of sign vectors of all elements in T and Σ(T ∗)=σ−1(σ(T \{0})) denotes the set of all vectors with the same sign of some nonzero vector in T . Theorem 5.2 ([65]). The following statements are equivalent: ∈ Rr (inj) The map fκ is injective with respect to S, for all κ >0. (sig) σ(ker(A)) ∩ σ(B(Σ(S∗)) = ∅.

BIOCHEMICAL REACTION NETWORKS 79

In the particular case m = n with rank(A)=rank(B)=n and S = Rn, condition (sig) simply reads σ(ker(A)) ∩ σ(im(B)) = {0}. In oriented matroid language [70], this can be phrased as: no nonzero vector of A is orthogonal to all vectors of BT , or, equivalently, no nonzero covector of BT is orthogonal to all covectors of A.In this framework, we recognized in [65] the first partial version of Descartes’ rule of signs, proposed by René Descartes in 1637 in “La Géometrie”, an appendix to his “Discours de la Méthode”. No multivariate generalization is known and only a lower bound together with a disproven conjecture was proposed in [56]. Recall that Descartes’ rule of signs says that given a univariate real polynomial r j f(x)=a0 + j=1 aj x , the number of positive real roots of f is bounded above by the number nf of sign variations in the ordered sequence of coefficient signs σ(a0),...,σ(ar) (where we discard the 0’s in this sequence and we add a 1 each time 6 8 111 two consecutive signs are different). For instance, if f = a0+3x−90x +2x +x , the sequence of coefficient signs (discarding 0’s) is: σ(a0), +, −, +, +. So, nf equals 2ifa0 ≥ 0and3ifa0 < 0. Then, f has at most 2 or 3 positive real roots. This bound is true in case of real, not necessarily natural, exponents. Note that being a condition only depending on the sign of the coefficients, the consequence should also hold for any other polynomial with the same vector of signs, that is, for r j any polynomial of the form fκ(x)=a0 + j=1 aj κj x , for any choice of positive ∈ Rr κ >0. The partial multivariate generalization is as follows. Given matrices A ∈ Rn×r, B ∈ Rr×n with n ≤ r and any index set J ⊆{1,...,r} of cardinality n,wedenote by det(AJ )(resp.det(BJ )) the minor indexed by the columns (resp. rows) in J. The following result from [65] was previously found but it was hidden in [14]. Theorem 5.3. [Multivariate Descartes’ bound for one positive root] Let A ∈ Rn×r, B ∈ Rr×n matrices of rank n. Assume that for all index sets J ⊆ [r] of cardinality n, the product of maximal minors det(AJ )det(BJ ) either is zero or has the same sign as all other non-zero such products, and moreover, at least one such product is non-zero. n Then, for any choice of (c1,...,cn) ∈ R , the system of equations r bj1 bjn (11) aij x1 ...xn = ci,i=1,...,n, j=1 ∈ Rn has at most one positive solution x >0. In particular, if the associated oriented matroids of A and B are equal, there is at most one positive solution. Note that in case n = 1, the conditions in Theo- r j ∈ R × rem 5.3 read as follows. For f = a0 + j=1 aj x [x], A is the 1 r matrix with entries a1,a2,...ar, B is the r×1 matrix with bj1 = j for all j =1,...,r, c1 = −a0. The hypotheses of the theorem reduce to asking that a1,...,ar ≥ 0(or≤ 0) and not all 0. So, there is at most one change sign (depending on σ(a0)) and so at most one positive root, as in classical Descartes’ rule. Indeed, Descartes’ rule ensures the existence of one positive root. For a multivariate version, see Corollary 3.13 in [65], based on Theorem 3.8 in [64].

6. Acknowledgments I am grateful to the organization of the First Mathematical Congress of the Americas for the invitation to speak at this wonderful event and to the editors of

80 ALICIA DICKENSTEIN this Proceedings Volume. I am also grateful to Murad Banaji, Carsten Conradi and Mercedes Pérez Millán for allowing me to use their figures. I am very thankful to my friends Reinhard Laubenbacher and Bernd Sturmfels, who opened for me the window to biological applications from an algebro-geometric background.

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Dipartimento de Matematica,´ FCEN - Universidad de Buenos Aires, and IMAS (UBA- CONICET), Ciudad Universitaria - Pab. I - C1428EGA Buenos Aires, Argentina E-mail address: [email protected] URL: http://mate.dm.uba.ar/~alidick

Contemporary Mathematics Volume 656, 2016 http://dx.doi.org/10.1090/conm/656/13078

Long-time asymptotic expansions for nonlinear diﬀusions in Euclidean space

Jochen Denzler, Herbert Koch, and Robert J. McCann

Abstract. We give a brief introduction to some recent developments concerning the long-time asymptotics of the porous medium and fast diﬀusion equations, focusing in particular on results contained in a recent monograph by the authors which rigorously relate the higher asymptotics of the nonlinear dynamics to its linearized spectrum for the latter equation. The statements and techniques are motivated using a comparison to the more familiar situation of the evolution towards Gaussian produced by the standard heat ﬂow.

The purpose of this announcement is to describe a few recent advances [12][24] in our understanding of the long-time behavior of the nonlinear diffusion equation ∂ρ (1) = ∇·(ρm−1∇ρ), ∂τ which governs the evolution of a density ρ(τ,·) ≥ 0onRn.Form = 1 this dynamics generalizes the linear heat equation to the case in which the thermal conductivity (or diffusion coefficient) is given by a power ρm−1 of the diffusing density. It can also be viewed as a scalar conservation law ∂ρ ρm−1 (2) = ∇·(ρ∇( )) ∂τ m − 1 1 m−1 in which the density ρ is advected by the gradient of the pressure m−1 ρ .The ranges m>1andm<1areknownastheporous medium and fast diffusion regimes respectively, depending on whether the rate of diffusion (or pressure) varies directly or inversely with density; their phenomenology, history and motivating applications are described in the book of Vázquez [26]. The advances described hereafter involve understanding the long-time behavior of solutions starting from integrable initial data of sufficiently rapid decay; to fix ideas we shall call the initial profile

(3) ρ0(·) = lim ρ(τ,·) τ→0

2010 Mathematics Subject Classiﬁcation. Primary 35B40. RJM acknowledges partial support of his research by Natural Sciences and Engineering Re- search Council of Canada Grant 217006-08. This material is based in part upon work supported by the National Science Foundation under Grant No. 0932078 000, while RJM was in residence at the Mathematical Science Research Institute in Berkeley, California, during the fall semester of 2014.

c 2016 American Mathematical Society 85

86 JOCHEN DENZLER, HERBERT KOCH, AND ROBERT J. MCCANN nice if it is integrable, non-negative, and compactly supported; the sense in which this limit holds needs to be made precise by specifying an appropriate topology. We are especially interested in the rates at which the dynamics causes diﬀerent aspects of the initial proﬁle to be dissipated / suppressed / forgotten. To understand what is possible in this direction, let us begin by recalling the familiar situation for the linear heat equation on Rn. There a well-known conjugacy to the quantum harmonic oscillator yields an expansion (6) to all orders which describes the decay of the various modes, as we now recall; c.f. Bartier et al [3] and the references there.

1. Long-time asymptotics for the heat equation on Rn

∂ρ n Fourier transforming the heat equation ∂τ =Δρ on R yields an exact formula 2 ρˆ(τ,k)=ˆρ(0,k)e−|k| τ for the rate of decay of the k-th Fourier mode 1 ik·x ρˆ(τ,k)= n/2 e ρ(τ,x)dx. (2π) Rn Only the zeroth Fourier mode fails to decay — since net mass is invariant under the heat flow. This description reflects the fact that nice initial data decay to zero under the heat flow, in any Lp(Rn) norm with p>1. However, this description misses many of the salient aspects of the evolution which are apparent either from its description in terms of Brownian motion, or from its explicit solution, expressed as a convolution of the initial data with the heat kernel: 1 2 ρ(τ,y)= ρ (z)e−|y−z| /4τ dz. n/2 0 (4πτ) Rn Either perspective shows that mass spreads in all directions from its initial location a distance proportional to τ 1/2 in time τ, and moreover that the shape of this spreading mass will necessarily become more and more Gaussian as time evolves, and details of the initial data are averaged away. It is the rate of this averaging away that we are interested in quantifying. To do so, let us renormalize the flow by setting 1 y (4) ρ(τ,y)= u(log τ, ). τ n/2 τ 1/2 Changing dependent variables from ρ to u corresponds to viewing the evolving mass distribution from a receding perspective: at each instant in time, the density 1 n u(log τ,·) has the same L (R )massasρ0, and corresponds to the density ρ(τ,·) viewed from distance τ 1/2. A standard computation $ % ∂ρ − n+2 n ∂u 1 − Δρ = τ 2 − u + − x ·∇u − Δu . ∂τ 2 ∂t 2 (t,x)=(log τ,y/τ1/2) shows ρ to be a solution of the heat equation if and only if ∂u 1 =Δu + ∇·(xu)=:−Lu. ∂t 2 −x2/4 This evolution fixes the Gaussian u(t, x)=e =: u∞(x), corresponding to a self-similar solution of the original dynamics, which is proportional to the heat 2 − n − y 1/2 kernel: ρ(τ,y)=τ 2 e 4τ .Thevariables(t, x)=(logτ,y/τ ) are sometimes called self-similar coordinates.

EXPANSIONS FOR NONLINEAR DIFFUSIONS IN EUCLIDEAN SPACE 87

Unlike the generator −Δ of the original dynamics, the operator L is not self- 2 n 2 n −1 n adjoint on L (R ), though it is self-adjoint on the weighted space L (R ,u∞ d x). −θ Notice the related quantity vθ(t, x)=u∞ (x)u(t, x) evolves according to a dynamics −θ θ generated by Lθ := u∞ Lu∞,namely ∂v − θ = L v ∂t θ θ − − 1 ·∇ − − n − |x|2 (5) = Δvθ +(θ 2 )x vθ (1 θ) 2 vθ + θ(1 θ) 4 vθ.

Choosing θ =1,weseetheevolutionoftherelativedensityv1 = u/u∞ is generated −1 2 n −|x|2/4 by a self-adjoint operator L1 = u∞ Hu∞ on the weighted space L (R ,e dx) 1 as in [3]. More remarkably, choosing θ = 2 we see the dynamics of v1/2 is generated by − − n 1 | |2 L1/2v = Δv 4 v + 16 x v 2 n which acts self-adjointly on the unweighted L (R ). Notice that L1/2 is essentially the Hamiltonian of the quantum harmonic oscillator, whose spectrum σ(L1/2)is well-known to consist of the non-negative integers and half-integers: σ(L1/2)= { 1 3 } 0, 2 , 1, 2 ,... .Fork =(k1,...,kn) with non-negative integer components, the n 1 normalized eigenfunction corresponding to eigenvalue λk := 2 ki is &n 1 2 ψ (x)= e−|x| /8 H (x /2) k (4π)n/4 ki i i=1

− k 2 k 2 √( 1) x d −x where Hk(x)= e k (e )isthek-th Hermite polynomial. Thus we can 2kk! dx 2 n −λ t expand v (t, x)= c (t)ψ (x)inL (R ), where c (t)=e k c(0) and 1/2 k k k

ck(0) = ψk(x)v1/2(0,x)dx Rn |x|2/8 = ψk(x)u(0,x)e dx. Rn Equivalently, ' ' ' ' ' n ' ' | |2 − ' − (6) 'e x /8u(t, x) − c (0)e t ki/2ψ (x)' ≤ Ce Λt ' k k ' ' n ' {0≤k ∈N| 1 k <Λ} i 2 i L2(Rn) 2 1/2 |x|2/8 −1/2 →∞ ≤ ≤ 2 n ∞ as t ,whereC ( ck(0) ) u(0,x)e L (R ).Thefactoru ≤ Λ λk multiplying the solution u(t, x) is reciprocal to the Gaussian factor in the eigenfunctions and suggests the convenience of expressing the convergence in appropriately weighted spaces; also, additional eigenfunctions with known coeﬃcients lead to faster and faster rates of decay.

2. Nonlinear diffusion If one is interested in the effects produced by a density dependent rate ρm−1 of diffusion (1), it is natural to wonder whether there is a description of the long-time behavior of this nonlinear evolution analogous to the linear case, in spite of the fact that the available tools for investigating the nonlinear problem must necessarily be quite different.

88 JOCHEN DENZLER, HERBERT KOCH, AND ROBERT J. MCCANN

− 2 Since the behavior depends crucially on the exponent m,letussetmp =1 n+p , where p is moment index introduced in [13].Threedistinctrangesofinterestare: − 2 the porous medium regime m>m∞ = 1, the extinction regime m

$ % 1 $ %− n+p − m−1 | |2 2 1 m 2 y uB(y):= B + |y| = B + 2 + n + p + where [λ]+ =max{λ, 0}.HereB>0 is a positive constant used to adjust the mass of the solution, which is finite for m>m0. The behavior manifested by these solutions varies across the three regimes mentioned above. In the porous medium regime it is a classical solution where positive, but because the rate of diffusion slows down where the density is small, the property of having compact support is preserved by the flow, and one has to understand the equation at the free boundary where ρ vanishes as prescribing that the free boundary move with a velocity given by the gradient of the pressure ρm−1/(m − 1), which is consistent with the conservation law (2) and typically incorporated into a suitable definition of weak solution. In the fast diffusion regime on the other hand, the rate of diffusion diverges at low densities, so that compactly supported initial data instantaneously develop thick tails whose moments are finite only up to order p; the BPKZ solution is a classical solution for t>0, which has finite mass if m>m0, and infinite mass otherwise. Clearly the BPKZ solutions are poor models for the behavior of nice initial data under the flow in the range mm0 on the other hand, it has been known since the work 1 of Friedman and Kamin [16]thatρB acts as an global attractor in L for the flow starting from nice initial data: ρ(τ,·) − ρB(τ,·)1 = o(1) as τ →∞,whereB is chosen so the initial mass of the solutions being compared coincides. For the one-dimensional porous medium equation n =1

EXPANSIONS FOR NONLINEAR DIFFUSIONS IN EUCLIDEAN SPACE 89

−β ρB(τ,·)1 = O(τ ). Otto’s method for doing this has proved particularly in- fluential. Rescaling the solution 1 y (7) ρ(τ,y)= u(log τ, ). τ nβ τ β in analogy with the linear case (4), he was able to show the rescaled dynamics ∂u 1 1 (8) = Δ(um)+ ∇·(xu) ∂t m 2 to be the gradient flow of an entropy 2 m 1 | |2 E(u)= − u (x)dx + u(x) x dx m(m 1) Rn 2 Rn with respect to the 2-Wasserstein distance d (u, u˜)2 =inf |x − y|2dγ(x, y). 2 ∈ γ Γ Rn×Rn This infimum is taken over all joint measures γ ≥ 0onRn × Rn with marginals u andu ˜ respectively. Obviously d2 =+∞ unless u has the same mass ofu ˜.Note the Barenblatt profile uB minimizes E(u) among densities u with fixed mass. For m ≥ mn, the entropy was known to be convex along 2-Wasserstein geodesics since McCann [19]; its modulus of convexity translates into a sharp rate of d2 contraction produced by the flow, which through suitable analysis can be converted into an L1 rate of convergence [21]. These analyses inspired various developments. On the one hand, Otto’s gradient flow formulation suggested that the linearization of the rescaled dynamics around the fixed profile uB would be governed by the Hessian of E(u)atuB.This Hessian acts self-adjointly on the tangent space to the set of probability measures, 1,2 n metrized by the weighted Hilbert space norm W (R ,uB). According to Be- namou and Brenier, this norm plays the role of a metric tensor generating the 2-Wasserstein distance [4]. In the fast diffusion regime m<1, the spectrum of this Hessian was computed by Denzler and McCann [13][14]. It consists of a finite number of eigenvalues +2k +(m − 1)(2 +2k + n − 2)k λ = k 2+n(m − 1) 1 (9) = [( +2k)p + n +4k(1 − − k)] 2p plus a semi-infinite interval of continuous spectrum beginning at 1 [(1 − m)(1 − n )+1]2 1 p (10) λcts = 2 = ( +1)2. 0 2 − n(1 − m) 2(1 − m) 2p 2 Here , k ∈ N are non-negative integers the corresponding eigenfunctions are poly- p nomials of degree +2k< 2 + 1 — just small enough to lie in the weighted space 1,2 n W (R ,uB). The multiplicity of λk coincides with the multiplicity of the -th n−1 spherical harmonic on S except at eigenvalue crossings (where λk = λk with (, k) =( ,k )). The lowest lying eigenvalues λ01 and λ10 correspond to translations in time and space, which commute with the flow (1); the next higher eigenvalue λ20 corresponds to affine shears, which do not.

90 JOCHEN DENZLER, HERBERT KOCH, AND ROBERT J. MCCANN

Concerning the nonlinear problem, it was shown that the L1 rate of convergence can be improved to O(τ −1) for initial data which is radially symmetric [9](byCar- rillo and Vázquez) or at least has its center of mass at the origin [20](byMcCann and Slepcev); the faster rate turns out to extend all the way to the threshold of the extinction regime m>m0 in these cases; see Carrillo and Vázquez for the radial case [9], Kim and McCann for the case m ∈ ]m0,m2][17] and Bonforte, Dolbeault, Grillo and Vázquez [6] or Denzler, Koch and McCann [12] for the general case. Sharp rates of convergence in entropy and L1 senses were eventually found in the full range of m by Blanchet, Bonforte, Dolbeault, Grillo and Vazquez [5]. In the fast-diffusion regime m ∈ ]m0, 1[, Vazquez also observed that convergence occurs in a stronger topology: the ratio of any two solutions tends to a constant in L∞(Rn), at a rate which has subsequently quantified by various groups of the authors above [9][17][12][6]. Although a further improvement becomes possible by centering the data in time as well as in space [15], what has remained elusive is a statement analogous to (6). Very recently, Christian Seis diagonalized the Hessian D2E(u) in the porous medium regime m>1. In contrast to the fast diffusion setting [14], which is plagued by the presence of continuous spectrum (9)–(10), he obtains a complete basis of eigenfunctions. However, it remains to be seen whether his diagonalization can be married to Koch’s framework [18] to produce a description of porous medium asymptotics in higher dimensions analogous to Angenent’s results on the line [1]. In the present manuscript we describe how such a marriage has been accomplished in the fast diffusion regime m ∈ ]m0, 1[ by Denzler, Koch and McCann [12]. Note added in proof: Since the submission of this work, the analogous marriage in the porous medium regime has been addressed through the development of an invariant manifold theory by Seis [23].

3. A dynamical systems approach Departing for a moment from the (infinite-dimensional) PDE setting, let us review what we are trying to achieve in the context of a (finite-dimensional) ODE setting. If we are interested in the long-time behavior of the initial value problem n x (t)=−V (x(t)) ∈ R with x(0) = x0, we can linearize the flow near near each fixed point V (x∞)=0: 2 (11) (x(t) − x∞) = −DV (x∞)(x(t) − x∞) + O(x(t) − x∞) ; the eigenvalues of DV (x∞) then determine the flow behavior nearby. If, in addition, the vector field V (x)=DE(x) has a gradient structure, then DV (x)=D2E(x)is 2 a symmetric matrix and its eigenvalues are real; denote them by σ(DE (x∞)) = {λ1 ≤ λ2 ≤ ...λn}. Then it is natural to expect n −λit 2λ1t x(t) − x∞ = cie + O(e ), i=1 2 which is in fact what happens unless the resonance 2λ1 ∈ σ(D E(x∞)) occurs between the linear and quadratic terms in (11), in which case the error term might be larger by a polynomial factor in t. Notice however, that this heuristic requires differentiable dependence of the vector field V (x) or equivalently of the flow X(t; x0) on its initial condition x0,atleastnearx∞. In the PDE context, this will mean

EXPANSIONS FOR NONLINEAR DIFFUSIONS IN EUCLIDEAN SPACE 91 we will need a well-posedness result which guarantees diﬀerentiable (as opposed to continuous) dependence on initial conditions.

4. The result The strategy of [12] is to adapt the finite-dimensional procedure caricatured above to the infinite-dimensional evolution of interest. The first challenge is to identify functional spaces in which the nonlinearity of the problem can be controlled, to yield a well-posedness result which includes differentiable dependence of the flow on initial conditions. This requires confronting — among other things — the degenerate parabolicity of the equation (8). Moreover, it turns out that the spaces in which this can be achieved are quite different from the spaces in which the linearized problem diagonalizes, a mismatch which must be reconciled. Finally, the possibilities of eigenvalue resonances and continuous spectrum must be addressed. As a sample of the results obtained: let us restrict our attention to initial conditions with center of mass at the origin so the low lying mode λ10 is not excited; the lowest remaining mode is then λ01. Fix a desired rate Λ of exponential decay as in (6). To avoid resonances and continuous spectra, assume Λ lies in the ∈ cts interval 2λ01 > Λ [λ01,λ0 ]. Theorem 4.1 (Fast diffusion asymptotics in weighted spaces). Fix p =2(1− −1 − ∈ cts m) n>2 and 2λ01 > Λ [λ01,λ0 ]. There is a sequence of polynomials { } ∈ p φk(x) —withφk(x) having degree +2k ]1, 2 +1[ — such that: For each solution u(t, x) with integrable, compactly supported initial data u0 and center of mass at the origin, there are coefficients ck such that ' ' − ' u(t,x) − − 1 λkt ' [ 1] | |2 ckφk(x)e ' uB (x) x ' ' B+ n+p ' ' 0<λk<Λ ' −Λt (12) ' √ ' = O(e ) ' |x|2 p−2− (p+2)2−4Λ /4 ' ' (B + n+p ) ' ∞ as t →∞, where the sum is over non-negative integers k, ∈ N for which λk = n − − ≤ (1 + p )/2+k +2(1 k)k/p lies in the interval ]0, Λ[ (and for which 1 if n =1). Let us remark on several aspects of this result beyond its resemblance to (6). ∞ n Here the space L (R ) satisfies the algebra property fg∞ ≤f∞g∞ which is relevant for controlling nonlinear corrections. The degree +2k polynomials φk(x) 2 2 |x| are the eigenfunctions of D E(uB); even after division by B + n+p they cannot lie in unweighted L∞ unless +2k<2. Thus the more terms which appear in the sum approximating u/uB, the more severely the weighted norm must discount growth at infinity to ensure the sum remains in the space. The ck represent the amplitudes of each excited mode in the range ]0, Λ[. One may naturally wonder how many distinct modes fall into this range? For appropriate choices of m and λ the answer can be as many as eight; it is possible to access even more modes by translating u in time to ensure the mode λ01 is not excited [12]. In contrast to the linear case, it is not possible to read the amplitudes ck off the initial data in any obvious way except when k = 0; in this case the eigenfunction φ0(x)isaharmonic polynomial, whose integral against the solution is therefore a conserved quantity of m the original flow mρτ =Δρ .

92 JOCHEN DENZLER, HERBERT KOCH, AND ROBERT J. MCCANN

5. A few ideas from the proof While we do not attempt even to sketch a proof here, we can never the less mention a few of its key ingredients. Since the solution u(t, x) decays to zero at spatial infinity, it does not stay a uniform distance from the singularity at zero of the nonlinearity u → um.Toovercome this lack of smoothness, we reexpress the dynamics in terms of the relative density v(t, x):=u(t, x)/uB(x); unlike the density, the relative density stays bounded above and below according to maximum principle type arguments of Vázquez; it tends uniformly to the constant 1 for an appropriate choice of B [25]. The relative density satisfies an evolution equation whose second-order term 1 |x|2 (13) v = ∇·[(B + )∇vm]+l.o.t.(Dv, v, |x|,B). t m n + p appears degenerate parabolic as |x|→∞. To cure this degenerate parabolicity of the dynamics linearized at v(t, x)=1,weviewRn as a (conformally flat) Riemann- ian manifold (M,g) with the so-called cigar metric

1 n ds2 = (dx )2, |x|2 i B + n+p i=1 introduced to this context independently by [7]and[12]. The second-order term m in the dynamics (13) is then given by the Laplace-Beltrami operator Δ(M,g)v /m. This allows us to combine DeGiorgi-Nash-Moser regularity with the implicit func- ∈ k,α ∩ L∞ −→ ∈ tion theorem to get differentiability of the flow v0 C (M) B (1) v Ck,α([0, ∞[×M) with respect to appropriate Hölder norms on the cigar — at least in a small uniform neighborhood of the fixed point v∞ =1. The linearized dynamics (v − 1)t = −L(v − 1) + o(v − 1) are generated by an operator L : Ck,α(M) −→ Ck,α(M) given in the coordinates ds = dr where r2 B+ n+p r = |x| by an expression like

−θ θ Lθ =(coshs) ◦ L ◦ (cosh s)

p ∂ p 2 = −Δ 2 +2( − 1 − θ)tanhs +( +1) (M,ds ) 2 ∂s 2 − p − − 2 − n p 2 − n p − − 2 1 ( 2 1 θ) (( 2 + 2 +1) ( 2 + 2 1 θ) ) cosh2 s . p − Here θ is selecting the strength of the weight, as in (5). Choosing θ = θcr := 2 1 suppresses the drift term, reducing Lθ to a Schr¨odinger operator on the cigar 2 | manifold with a universal potential. This operator is related to H = D E uB : 1,2 n −→ 1,2 n ◦ ◦ W (R ,uB) W (R ,uB) through conjugation Lθcr Λ=Λ H by the dif- 1 ∇· ∇ 1 ◦ ferential operator Λφ = (uB φ)= | |2 H and also by the multiplication uB x B+ n+p ◦ 1 1 ◦ operator Lθcr |x|2 = |x|2 H.Here B+ n+p B+ n+p |x|2 Hφ = −(B + )Δ n +(p + n)x ·∇φ n + p R is the operator diagonalized by Denzler and McCann [13][14]ands is geodesic distance along the cigar.

EXPANSIONS FOR NONLINEAR DIFFUSIONS IN EUCLIDEAN SPACE 93

The decay rate Λ of the error term in (12) determines the relevant choice of θ = θcr. Thus we actually work in weighted H¨older spaces on the cigar, but the weighted H¨older norms also control weighted L∞.

References [1] Sigurd Angenent, Local existence and regularity for a class of degenerate parabolic equations, Math. Ann. 280 (1988), no. 3, 465–482, DOI 10.1007/BF01456337. MR936323 (89e:35072) [2] G. I. Barenblatt, On some unsteady motions of a liquid and gas in a porous medium (Russian), Akad. Nauk SSSR. Prikl. Mat. Meh. 16 (1952), 67–78. MR0046217 (13,700a) [3] Jean-Philippe Bartier, Adrien Blanchet, Jean Dolbeault, and Miguel Escobedo, Improved intermediate asymptotics for the heat equation, Appl. Math. Lett. 24 (2011), no. 1, 76–81, DOI 10.1016/j.aml.2010.08.020. MR2727993 (2011i:35100) [4] Jean-David Benamou and Yann Brenier, A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem,Numer.Math.84 (2000), no. 3, 375–393, DOI 10.1007/s002110050002. MR1738163 (2000m:65111) [5] Adrien Blanchet, Matteo Bonforte, Jean Dolbeault, Gabriele Grillo, and Juan Luis Vázquez, Asymptotics of the fast diffusion equation via entropy estimates, Arch. Ration. Mech. Anal. 191 (2009), no. 2, 347–385, DOI 10.1007/s00205-008-0155-z. MR2481073 (2011d:35236) [6] M. Bonforte, J. Dolbeault, G. Grillo, and J. L. Vázquez, Sharp rates of decay of solutions to the nonlinear fast diffusion equation via functional inequalities, Proc. Natl. Acad. Sci. USA 107 (2010), no. 38, 16459–16464, DOI 10.1073/pnas.1003972107. MR2726546 (2011g:35187) [7] Matteo Bonforte, Gabriele Grillo, and Juan Luis Vázquez, Special fast diffusion with slow asymptotics: entropy method and flow on a Riemann manifold,Arch.Ration.Mech.Anal. 196 (2010), no. 2, 631–680, DOI 10.1007/s00205-009-0252-7. MR2609957 (2011d:35259) [8] J. A. Carrillo and G. Toscani, Asymptotic L1-decay of solutions of the porous medium equation to self-similarity, Indiana Univ. Math. J. 49 (2000), no. 1, 113–142, DOI 10.1512/iumj.2000.49.1756. MR1777035 (2001j:35155) [9] José A. Carrillo and Juan L. Vázquez, Fine asymptotics for fast diffusion equations, Comm. Partial Differential Equations 28 (2003), no. 5-6, 1023–1056, DOI 10.1081/PDE-120021185. MR1986060 (2004a:35118) [10] P. Daskalopoulos and N. Sesum, Eternal solutions to the Ricci flow on R2,Int.Math.Res. Not., posted on 2006, Art. ID 83610, 20, DOI 10.1155/IMRN/2006/83610. MR2264733 (2007f:53078) [11] Manuel Del Pino and Jean Dolbeault, Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diffusions (English, with English and French summaries), J. Math. Pures Appl. (9) 81 (2002), no. 9, 847–875, DOI 10.1016/S0021-7824(02)01266-7. MR1940370 (2003h:35051) [12] J. Denzler, H. Koch, and R.J. McCann. Higher-order time asymptotics of fast diffusion in euclidean space (via dynamical systems methods). Mem. Amer. Math. Soc. 234:1–94, 2015. [13] Jochen Denzler and Robert J. McCann, Phase transitions and symmetry breaking in singular diffusion,Proc.Natl.Acad.Sci.USA100 (2003), no. 12, 6922–6925 (electronic), DOI 10.1073/pnas.1231896100. MR1982656 (2004c:35210) [14] Jochen Denzler and Robert J. McCann, Fast diffusion to self-similarity: complete spectrum, long-time asymptotics, and numerology, Arch. Ration. Mech. Anal. 175 (2005), no. 3, 301– 342, DOI 10.1007/s00205-004-0336-3. MR2126633 (2005k:35214) [15] Jean Dolbeault and Giuseppe Toscani, Fast diffusion equations: matching large time asymptotics by relative entropy methods, Kinet. Relat. Models 4 (2011), no. 3, 701–716, DOI 10.3934/krm.2011.4.701. MR2823993 [16] Avner Friedman and Shoshana Kamin, The asymptotic behavior of gas in an n-dimensional porous medium, Trans. Amer. Math. Soc. 262 (1980), no. 2, 551–563, DOI 10.2307/1999846. MR586735 (81j:35054) [17] Yong Jung Kim and Robert J. McCann, Sharp decay rates for the fastest conservative diffusions (English, with English and French summaries), C. R. Math. Acad. Sci. Paris 341 (2005), no. 3, 157–162, DOI 10.1016/j.crma.2005.06.025. MR2158837 (2006b:35170) [18] H. Koch. Non-Euclidean Singular Integrals and the Porous Medium Equation. 1999. Habili- tation Thesis, Unversität Heidelberg, Germany.

94 JOCHEN DENZLER, HERBERT KOCH, AND ROBERT J. MCCANN

[19] Robert J. McCann, A convexity principle for interacting gases, Adv. Math. 128 (1997), no. 1, 153–179, DOI 10.1006/aima.1997.1634. MR1451422 (98e:82003) [20] Robert J. McCann and Dejan Slepˇcev, Second-order asymptotics for the fast- diffusion equation, Int. Math. Res. Not., posted on 2006, Art. ID 24947, 22, DOI 10.1155/IMRN/2006/24947. MR2211152 (2006k:35150) [21] Felix Otto, The geometry of dissipative evolution equations: the porous medium equation, Comm. Partial Differential Equations 26 (2001), no. 1-2, 101–174, DOI 10.1081/PDE- 100002243. MR1842429 (2002j:35180) [22] R. E. Pattle, Diffusion from an instantaneous point source with a concentration-dependent coefficient, Quart. J. Mech. Appl. Math. 12 (1959), 407–409. MR0114505 (22 #5326) [23] C. Seis. Invariant manifolds for the porous medium equation. Preprint at arXiv:1505.06657. [24] Christian Seis, Long-time asymptotics for the porous medium equation: the spectrum of the linearized operator, J. Differential Equations 256 (2014), no. 3, 1191–1223, DOI 10.1016/j.jde.2013.10.013. MR3128937 [25] Juan Luis Vázquez, Asymptotic beahviour for the porous medium equation posed in the whole space,J.Evol.Equ.3 (2003), no. 1, 67–118, DOI 10.1007/s000280300004. Dedicated to Philippe Bénilan. MR1977429 (2004d:35138) [26] Juan Luis Vázquez, The porous medium equation, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2007. Mathematical theory. MR2286292 (2008e:35003) [27] Ya.B. Zel’dovich and G.I. Barenblatt. The asymptotic properties of self-modelling solutions of the nonstationary gas filtration equations. Sov. Phys. Doklady, 3:44–47, 1958. [28] Ya.B. Zel’dovich and A.S. Kompaneets. Theory of heat transfer with temperature dependent thermal conductivity. In Collection in Honour of the 70th Birthday of Academician A.F. Ioffe, pages 61–71. Izdvo. Akad. Nauk. SSSR, Moscow, 1950.

Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996 E-mail address: [email protected] Mathematisches Institut, Universitat¨ Bonn, Endenischer Allee 60, 53115 Bonn, Germany E-mail address: [email protected] Department of Mathematics, University of Toronto, Toronto Ontario M5S 2E4 Canada E-mail address: [email protected]

Contemporary Mathematics Volume 656, 2016 http://dx.doi.org/10.1090/conm/656/13104

Non-strongly isospectral spherical space forms

E.A.Lauret,R.J.Miatello,andJ.P.Rossetti

Abstract. In this paper we describe recent results on explicit construction of lens spaces that are not strongly isospectral, yet they are isospectral on p-forms for every p. Such examples cannot be obtained by the Sunada method. We also discuss related results, emphasizing on signiﬁcant classical work of Ikeda on isospectral lens spaces, via a thorough study of the associated generating functions.

1. Introduction Two compact Riemannian manifolds are said to be isospectral if the spectra of their Laplace operators on functions are the same. More generally, they are said to be p-isospectral if the spectra of their Hodge-Laplace operators acting on p-forms are the same. Recently, in [LMR15b], we have found examples of pairs of lens spaces that are p-isospectral for every p. Since lens spaces have cyclic fundamental group, they cannot be strongly isospectral. To the best of our knowledge these are the first (connected) examples of this kind. By showing a nice connection between isospectrality of lens spaces and isospectrality of certain associated integral lattices with respect to the one-norm, we were able to construct an infinite family of pairs of 5-dimensional lens spaces that are p-isospectral for every p. Before this, A. Ikeda found many interesting examples of isospectral lens spaces. The main tool of his approach was the generating function associated to the spectrum. Our method does not use generating functions, but relies on the representation theory of compact Lie groups. In view of our construction of new families and the opening connection with one-norm isospectral integral lattices, we expect it will be useful to write this article attempting to bring together in a more accessible way, our method, the foundational work of Ikeda and the method of Sunada. Historically, the first example of isospectral non-isometric manifolds was a pair of tori constructed by using lattices of dimension n =16([Mi64], [Wi41]). The dimension was reduced from 16 to 4 in several articles (see [Schi90], [CS92]and the references therein). Such lattices are isospectral with respect to the standard norm ·2, that is, for each length they have the same number of vectors of that length.

2010 Mathematics Subject Classiﬁcation. Primary 58J53. Key words and phrases. Isospectral, spherical space forms, lens spaces, p-spectrum.

c 2016 American Mathematical Society 95

96 E. A. LAURET, R. J. MIATELLO, AND J. P. ROSSETTI

Besides these examples, many other contributions have been given, showing different connections between the spectra and the geometry of a Riemannian manifold. In [Su85] T. Sunada gave a general method that produces strongly isospectral manifolds, that is, manifolds isospectral for every natural strongly elliptic operator acting on sections of a natural vector bundle, in particular they are p-isospectral for all p. Later, this method was extended and applied by many authors, in particular by D. DeTurck-C. Gordon [DG89], P. Bérard [Be93]andH.Pesce[Pe96]. In the context of spherical space forms, A. Ikeda [Ik83], P. Gilkey [Gi85], J. A. Wolf [Wo01] produced Sunada isospectral forms with non-cyclic fundamental groups. The construction of manifolds that are p-isospectral for some values of p only cannot be attained by Sunada’s method. The first such pair was given by C. Gordon in [Go86]. Among other known examples we mention those in [Gt00] for nilmanifolds and those given in [MR01], [MR03], [DR04] for compact flat manifolds. A. Ikeda studied the spectrum of spherical space forms in several interesting articles (see [Ik80a], [Ik80b],[Ik80c], [Ik83], [Ik88]). He developed the theory of generating functions associated to spectra, obtaining many isospectral examples of Sunada and non-Sunada type. In particular, for each given p0, he constructed families of lens spaces that are p-isospectral for every 0 ≤ p ≤ p0, but are not p0 +1- isospectral. None of Ikeda’s examples of isospectral lens spaces are p-isospectral for all p and actually until very recently, no examples were known of compact Riemann- ian manifolds that are p-isospectral for every p but are not strongly isospectral. This question has been around for some time (see [Wo01, p. 323]). In [LMR15b]we find a rather surprising two-parameter infinite family of pairs of lens spaces that are p-isospectral for every p, but are not strongly isospectral. We also give many more examples obtained with the help of the computer and also examples in arbitrarily large dimensions. The paper is organized as follows. Section 2 is devoted to describe summarily Sunada’s method and its generalizations. In Section 3 we develop the necessary tools of representation theory of compact Lie groups to be used in the proofs of our main results in Section §5. Section 4 is devoted to Ikeda’s important work, that is scattered in several papers that are sometimes hard to follow. We have tried to make it more accessible, including the main ideas in most of the proofs. In Section 5 we describe our construction of isospectral lens spaces in dimension n =2m − 1 by means of one-norm isospectral integral lattices in Zm. A detailed description of the methods and the results is given at the beginning of the section. The paper finishes with tables, obtained by computer methods, listing all existing examples for n = 5, 7 and 9, where the order of the fundamental group q is less than 500, 300 and 150 respectively. We have left some open questions or problems, usually at the end of the sections or subsections. Acknowledgement. The authors wish to thank Peter Doyle for stimulating discus- sions and for facilitating the use of fast computer programs to check the tables in Section 5.

2. Sunada’s method T. Sunada [Su85] gave a simple and effective method that allowed to produce a great variety of examples of isospectral manifolds. It is based on a triple of finite groups Γ1, Γ2,G,whereΓ1, Γ2 are subgroups of G that are almost conjugate in G, that is, there is a bijection from Γ1 to Γ2 that preserves G-conjugacy. The first

NON-STRONGLY SPHERICAL SPACE FORMS 97 such triples were given by Gassmann [Ga26] who used them to give pairs of nonisomorphic number ﬁelds having the same Dedekind zeta function. The Sunada theorem can be stated as follows.

Theorem 2.1. Let Γ1, Γ2 be almost conjugate subgroups of a ﬁnite group G. Assume that G acts by isometries on a Riemannian manifold M in such a way that Γ1, Γ2 act freely. Then the manifolds Γ1\M and Γ2\M are strongly isospectral.

Given a Gassmann triple Γ1, Γ2,G, to place oneself in the conditions of Sunada’s theorem it is suﬃcient to give a Riemannian manifold M0 such that there is a surjective homomorphism φ : π1(M0) → G. Sunada gave many applications of this theorem, in particular, he constructed large sets of pairwise isospectral non- isometric Riemann surfaces for any genus g ≥ 5. Also, he showed that manifolds Γ1\M and Γ2\M as in the theorem must have the same lengths of closed geodesics. We note, however, that these lengths need not have the same multiplicities (see for instance [Go85], [Gt94], [Gt96]and[MR03]). Sunada’s result was intensely exploited and was followed by several generalizations. Still today, the method accounts for most of the known examples of isospectral manifolds. We note that the condition of almost conjugacy in the ﬁnite group G is equivalent to a condition in terms of group representations, namely, that the right regular representations of G on the function spaces C(Γ1\G)and C(Γ2\G) are equivalent representations. More generally, if G is a Lie group and Γ1, Γ2 are discrete cocompact subgroups, then Γ1,Γ2 are said to be representation 2 2 equivalent in G, if the right regular representations of G on L (Γ1\G)andL (Γ2\G) are equivalent representations. The following generalization of Theorem 2.1, due to DeTurck-Gordon [DG89] (see also [Be93]), is very useful. Theorem 2.2. Let G be a Lie group acting by isometries on a Riemannian manifold M and let Γ1, Γ2 be discrete subgroups of G such that Γ1\M and Γ2\M are compact manifolds. If, furthermore, Γ1, Γ2 are representation equivalent in G, then Γ1\M and Γ2\M are strongly isospectral. One can give a convenient reformulation of the condition of representation equivalence in the theorem. Namely, if g ∈ G denote by C(g, Γi), C(g, G)the centralizers of g in Γi and G respectively. Under the conditions above, the quotient C(g, Γi)\C(g, G)iscompactfori =1, 2. One has that Γ1,Γ2 are representation equivalent in G if and only if, for each g ∈ G, (2.1) vol(C(a, Γ1)\C(a, G)) = vol(C(b, Γ2)\C(b, G)). ⊂ ⊂ [a]Γ1 [g]G [b]Γ2 [g]G

Here [g]G,[a]Γ1 and [b]Γ2 denote respectively the conjugacy classes of g in G,ofa in Γ1 and of b in Γ2 and the volumes are computed with respect to suitable invariant measures in C(g, G)andC(g, Γi). As a consequence, one obtains the following (see [Wo01])

Corollary 2.3. If G is a compact Lie group and the Γi are ﬁnite, then Γ1 and Γ2 are representation equivalent in G if and only if Γ1, Γ2 are almost conjugate in G. H. Pesce studied the relation between representation theory and isospectrality in several papers ([Pe95], [Pe96], [Pe98]). In particular, in [Pe95]heprovedthat

98 E. A. LAURET, R. J. MIATELLO, AND J. P. ROSSETTI the converse of the Sunada condition is satisfied for manifolds of curvature ±1. n n That is, if X = S or X = H , G = I(X)andΓ1 and Γ2 are discrete cocompact subgroups of G and if Γ1\X and Γ2\X are strongly isospectral, then Γ1 and Γ2 are representation equivalent in G.(InthecaseofX = Rn this result is proved in [La14].) Also, in a subsequent paper ([Pe96]), he gives a generalization of the condition of representation equivalence, by introducing the weaker notion of τ-representation equivalence,whereτ is a representation of the compact Lie group K that is the generic stabilizer of the action. When τ is the trivial representation he calls the discrete subgroups K-equivalent. As an application, in the case of spaces of constant curvature he shows that Γ1 and Γ2 are K-equivalent in G = I(X) if and only if Γ1\X and Γ2\X are 0-isospectral. A generalization was given recently in [LMR15a], in the same context, for X = G/K of constant curvature when τ = τp is the p-exterior representation of K =O(n). We showed that in the elliptic case, for each fixed p,Γ1 and Γ2 are τp-equivalent in G = I(X) if and only if Γ1\X and Γ2\X are p-isospectral. However, in the flat and hyperbolic cases, we prove that Γ1 and Γ2 are τq equivalent in G = I(X) for every 0 ≤ q ≤ p, if and only if Γ1\X and Γ2\X are q-isospectral for every 0 ≤ q ≤ p. Also we gave examples showing that in the flat case, p-isospectrality is far from implying τp-equivalence for each fixed p. To conclude this section, we list a number of representative papers illustrating the construction of strongly isospectral manifolds by means of the Sunada method or its generalizations. (i) Isospectral Riemann surfaces: [Vi80], [Su85], [BT87], [BGG98], [Br96], [Bu92], [Bu86], [GMW05]. (ii) Isospectral spherical space forms: [Ik83], [Gi85], [Wo01]. (iii) Isospectral locally symmetric: [Vi80], [Sp89], [McR06]. (iv) Continuous isospectral families: [GW84], [GW97], [Schu95]. (v) Isospectral graphs: [Bu88], [Br96], [FK99]. (vi) Isospectral planar domains: [GWW92], [Bu88], [BCDS94]. (vii) Isospectral flat manifolds: [DM92], [DR04], [MR99], [MR03], [LMR13]. (viii) τ-representation equivalent manifolds: [Pe96], [Pe98], [Su02], [LMR15a]. For a more complete discussion of the Sunada method we refer to the surveys by C. Gordon [Go09], [Go00]. In the remaining sections we will discuss several isospectrality situations in the case of spherical space forms, that are not of the strong type, thus they cannot be obtained by the Sunada method.

3. Spectra of spherical space forms In this section we will recall various facts on spectra of spherical space forms. We refer to [IT78] for the main basic facts. We will describe the results in the language of representation theory of orthogonal groups. The n-dimensional sphere Sn is a symmetric space realized as G/K with G =SO(n+1), K =SO(n). If Γ is a ﬁnite subset of SO(n+1) acting freely on Sn, then the manifold Γ\Sn is a spherical space form. We restrict our attention to the odd-dimensional case n =2m − 1, since the only manifold covered (properly) by S2m is P R2m.

NON-STRONGLY SPHERICAL SPACE FORMS 99

We consider the standard maximal torus T in SO(2m), with Lie algebra given by (3.1) h := H = diag 02πθ1 ,..., 02πθm : θ ∈ Rm . 0 −2πθ1 0 −2πθm 0 ∗ Its complexification is a Cartan subalgebra h of so(2m, C). As usual, define εj ∈ h − ≤ ≤ ∈ by εj (H)= 2πiθj for( any 1 j m, H h. The weight lattice of G is thus m Z given by P (SO(2 m)) = j=1 εj . We use the standard system of positive roots, m ≥···≥ ≥| | thus a weight j=1 aj εj is dominant if and only if a1 am−1 am . · · ∗ Let , be the inner product on ih0 so that ε1,...,εm is an orthonormal basis. This is the dual of the positive multiple of the Killing form that induces on S2m−1 the Riemannian metric with constant curvature equal to one. In K = {g ∈ SO(2m):ge2m = e2m}⊂SO(2m), we take the maximal torus TK = T ∩ K, thus the associated Cartan subalgebra hK can be seen as included in h in the usual way. Under this( convention, the weight lattice of K can be − m−1 Z m−1 identified with P (SO(2m 1)) = j=1 εj and j=1 aj εj is dominant if and only if a1 ≥···≥am ≥ 0. Let G) and K) denote respectively the equivalence classes of unitary irreducible representations of G and K respectively, endowed with invariant inner products. By the highest weight theorem, the elements in G) (resp. K)) are in a bijective correspondence with the dominant weights Λ of G (resp. μ of K). For each Λ, ) we denote by πΛ ∈ G the irreducible representation with highest weight Λ. For ∈ ) example, πkε1 G, with highest weight kε1, can be realized in the space of complex homogeneous harmonic polynomials of degree k,inm variables. ) For (τ,Wτ ) ∈ K,letEτ denote the associated homogeneous vector bundle × −→ 2m−1 2m−1 § 2 Eτ := G τ Wτ S of S (see [Wa73, 5.2]). The space of L -sections 2 ⊗ of Eτ decomposes as L (Eτ ) π∈G Vπ HomK (Vπ,Wτ ), where G acts in the first variable in the right-hand side. If Γ is a finite subgroup of G,thespaceΓ\Eτ is a vector bundle over Γ\S2m−1 with L2-sections given by the Γ-invariant elements 2 of L (Eτ ); thus we have the decomposition 2 \ 2 Γ Γ ⊗ (3.2) L (Γ Eτ )=L (Eτ ) Vπ HomK (Vπ,Wτ ). π∈G

The Laplace operator Δτ,Γ acting on smooth sections Γ\Eτ can be identiﬁed with the action of the Casimir element C ∈ U(so(2m, C)) (the universal enveloping C Γ ⊗ algebra of so(2m, )). On each summand Vπ HomK (Vπ,Wτ ), C acts by the scalarλ(C, π)=Λ+ρ, Λ+ρ−ρ, ρ, where Λ is the highest weight of π and m − ∈ R ρ = j=1(m j)εj . In particular, the multiplicity dλ(τ,Γ) of λ in the spectrum of Δτ,Γ equals Γ (3.3) dλ(τ,Γ) = dim Vπ [τ : π], π∈G: λ(C,π)=λ where [τ : π]=dim(HomK (Vπ,Wτ )) can be computed by the well known branching − ∈ ) law from G =SO(2m)toK =SO(2m 1). That is, if τ K has highest weight m−1 ∈ ) m μ = j=1 bj εj and π G has highest weight Λ = j=1 aj εj ,then[τ : π] > 0if and only if

(3.4) a1 ≥ b1 ≥ a2 ≥ b2 ≥···≥am−1 ≥ bm−1 ≥|am|.

100 E. A. LAURET, R. J. MIATELLO, AND J. P. ROSSETTI

) { ∈ ) } Moreover, the branching is multiplicity free. Hence, Gτ := π G :[τ : π]=1 is ∈ ) m the set of πΛ G (Λ = j=1 aj εj ) such that (3.4) holds. We can now describe the τ-spectrum of any spherical space form Γ\S2m−1. Theorem 3.1. Let Γ be a ﬁnite subgroup of G =SO(2m) and let τ be an irreducible representation of K =SO(2m − 1).Then,λ ∈ R is an eigenvalue of ) Δτ,Γ if and only if λ = λ(C, π) for some π ∈ Gτ . In this case, its multiplicity is given by Γ dλ(τ,Γ) = dim Vπ , ) the sum taken over π ∈ Gτ such that λ(C, π)=λ. If τ is an irreducible representation of K =SO(2m − 1), then the spaces Γ\S2m−1 and Γ\S2m−1 are said to be τ-isospectral if the Laplace type operators Δτ,Γ and Δτ,Γ have the same spectrum. If τp denotes the irreducible representation of SO(2m−1) with highest weight ε1 +···+εp for 0 ≤ p ≤ m−1, then the associated

Laplace operator Δτp,Γ can be identiﬁed with the Hodge-Laplace operator Δp acting 2m−1 on p-forms of Γ\S . As usual, we call p-spectrum the spectrum of Δp and we 2m−1 write p-isospectral in place of τp-isospectral. Since Γ ⊂ SO(2m), then Γ\S is always orientable, hence the p-spectrum and the 2m − 1 − p-spectrum are the same. We next restate Theorem 3.1 for τ = τp. We ﬁrst introduce some more notation. ··· ± ··· ± Let Λp = ε1 + + εp for p

Here 1 denotes the trivial representation π0 of SO(2m). We now set E0 = {0} and 2 (3.6) Ep = {λk := λ(C, πk,p)=k + k(2m − 2) + (p − 1)(2m − 1 − p):k ∈ N0} for 1 ≤ p ≤ m. Theorem 3.2. Let Γ be a ﬁnite subgroup of G =SO(2m) and let 0 ≤ p ≤ m−1. ∈ R ∈E ∪E If λ is an eigenvalue of Δτp,Γ then λ p p+1. Its multiplicity is given by dim V Γ if λ = λ ∈E , πk,p k p dλ(τp, Γ) = dim V Γ if λ = λ ∈E . πk,p+1 k p+1

In particular, when p =0, the eigenvalues of the Laplace-Beltrami operator Δτ0,Γ lie 2 Γ 2 in the set {k +k(2m−2) : k ∈ N0} and dλ(τ0, Γ) = dim V if λ = k +k(2m−2). πkε1

From Theorem 3.2 and the fact that Ep ∩Ep+1 = ∅ when p>0, we obtain the following characterizations. Corollary 3.3. Let Γ and Γ be ﬁnite subgroups of SO(2m). (i) Γ\S2m−1 and Γ\S2m−1 are 0-isospectral if and only if dim V Γ =dimV Γ πkε1 πkε1 for every k ∈ N.

NON-STRONGLY SPHERICAL SPACE FORMS 101

(ii) If 1 ≤ p ≤ m − 1, Γ\S2m−1 and Γ\S2m−1 are p-isospectral if and only if

dim V Γ =dimV Γ and dim V Γ =dimV Γ πk,p πk,p πk,p+1 πk,p+1 for every k ∈ N. (iii) Γ\S2m−1 and Γ\S2m−1 are p-isospectral for all p if and only if dim V Γ = πk,p dim V Γ for every k ∈ N and every 1 ≤ p ≤ m − 1. πk,p

4. The work of Ikeda In this section we will give a summary of Ikeda’s important work on isospectral spherical space forms. Our notation will somewhat diﬀer from Ikeda’s in that we use the language of representation theory introduced in the previous section. Generating functions are a main tool in Ikeda’s work. One can encode the 0-spectrum of a space Γ\S2m−1 in the function F 0(z)= dim V Γ zk. Γ πkε1 k≥1

In light of Corollary 3.3 (i), Γ\S2m−1 and Γ\S2m−1 are 0-isospectral if and only 0 0 0 if FΓ(z)=FΓ (z).Ikedaprovedin[Ik80b, Thm. 2.2] that FΓ(z)convergesfor |z| < 1 to the rational function 1 1 − z2 1 (4.1) F 0(z)= − . Γ |Γ| det(1 − zγ) |Γ| γ∈Γ * − − − Here det(1 zγ) stands for det(Id2m zγ)= λ(1 zλ), where λ runs over the eigenvalues of γ.Notethatdet(1− zγ)=det(z − γ) for any γ ∈ SO(2m), since for any λ an eigenvalue of γ one has |λ| =1andλ is also an eigenvalue. He observed (see [Ik80b, Corollary 2.3]) that (4.1) implies that if Γ and Γ are almost conjugate subgroups of SO(2m)thenΓ\S2m−1 and Γ\S2m−1 are 0- isospectral. This result can be viewed as a predecessor of Sunada’s method. In [Ik83], Ikeda constructed explicitly non-isometric isospectral spherical space forms by using this method. These pairs are always strongly isospectral and have non- cyclic fundamental group. P. Gilkey [Gi85] independently found very similar examples. Later, J. A. Wolf [Wo01] made a step in the determination of all strongly isospectral spherical space forms by using the classiﬁcation in [Wo67]. In what follows, we will focus our interest on isospectral spherical space forms that are not strongly isospectral. Ikeda in [Ik88] encoded, for any p ≥ 1, the p-spectrum of a spherical space form Γ\S2m−1 by means of generating functions. He deﬁned, as a generalization of 0 FΓ(z) the function (4.2) F p(z)= dim V Γ zk. Γ πk,p+1 k≥0

p Although FΓ (z) does not have information on all of the p-spectrum, by Theo- p p−1 rem 3.2, the p-spectrum is determined by FΓ (z)andFΓ (z) together. In particu- \ 2m−1 \ 2m−1 p−1 p−1 lar, Γ S and Γ S are p-isospectral if and only if FΓ (z)=FΓ (z)and p p FΓ (z)=FΓ (z) by Corollary 3.3 (ii).

102 E. A. LAURET, R. J. MIATELLO, AND J. P. ROSSETTI

He proved, by using a convenient realization of the representation Vπk,p ,the following neat formula (see [Ik88, p. 394]): p 1 χk(γ) (4.3) F p(z)=(−1)p+1z−p + (−1)p−k(zk−p − zp−k+2) . Γ |Γ| det(z − γ) k=0 γ∈Γ + Here, χk denotes the character of the k-exterior representation k(C2m)ofSO(2m). k k χ (γ) ≤ ≤ − Set FΓ (z)= γ∈Γ det(z−γ) for each 0 k m 1. As a direct consequence of (4.3) he obtains the following result ([Ik88, Prop. 2.4]).

Proposition 4.1. Let p0 ∈ Z, 0 ≤ p0 ≤ m − 1. Two spherical space forms \ 2m−1 \ 2m−1 ≤ ≤ p Γ S and Γ S are p-isospectral for all 0 p p0 if and only if FΓ (z)= p ≤ ≤ FΓ (z) for every 0 p p0. By using Proposition 4.1, Ikeda also was able to characterize spherical space forms that are p-isospectral for all p.Ifw is an indeterminate, one can check that 2m − k k k − k=0( 1) χ (γ) w =det(w γ), thus 2m det(w − γ) (4.4) Q (w, z):= (−1)k Fk(z) wk = . Γ Γ det(z − γ) k=0 γ∈Γ Therefore one has the following characterization (see [Ik88, Thm. 2.5]). Theorem 4.2. Two spherical space forms Γ\S2m−1 and Γ\S2m−1 are p-isospectral for all p if and only if QΓ(w, z)=QΓ (w, z). In a similar way as in our comment after (4.1), Theorem 4.2 implies that almost conjugate subgroups yield manifolds that are p-isospectral for all p (see [Ik88, Thm. 2.7]). The previous results are valid for generating functions of arbitrary spherical space forms. As an application, Ikeda proved the existence of many families of non- isometric 0-isospectral lens spaces. Since Pesce [Pe95] has proved that strongly isospectral lens spaces are necessarily isometric (see also [LMR15b, Prop. 7.2]), it turns out that these examples cannot be obtained by Sunada’s method. From now on we will focus on lens spaces, that is, spherical space forms with cyclic fundamental group. They can be described as follows. For each q ∈ N and s1,...,sm ∈ Z coprime to q,denote 2m−1 (4.5) L(q; s1,...,sm)=γ\S where ,- . - ./ (4.6) γ = diag cos(2πs1/q) sin(2πs1/q) ,..., cos(2πsm/q) sin(2πsm/q) − sin(2πs1/q)cos(2πs1/q) − sin(2πsm/q)cos(2πsm/q) The element γ generates a cyclic group of order q in SO(2m) that acts freely on S2m−1. The following fact is well known (see [Co70,Ch.V]). Proposition . 4.3 Let L = L(q; s1,...,sm) and L = L(q; s1,...,sm) be lens spaces. Then the following assertions are equivalent. (1) L is isometric to L. (2) L is diﬀeomorphic to L. (3) L is homeomorphic to L. m (4) There exist t ∈ Z coprime to q and ∈{±1} such that (s1,...,sm) is a permutation of (1ts1,...,mtsm)(modq).

NON-STRONGLY SPHERICAL SPACE FORMS 103

Furthermore, L and L are homotopically equivalent if and only if there exists t ∈ Z ≡±m such that s1 ...sm t s1 ...sm (mod q). 2m−1 Let L = L(q; s1,...,sm)=Γ\S be a lens space and let ξ =exp(2πi/q). From (4.1), one has that q − 2 0 1 * 1 z − (4.7) FΓ(z)= m − 1. q (z − ξsj l)(z − ξ sj l) l=1 j=1 This formula was first pointed out in [IY79,Thm.3.2]. We now sketch Ikeda’s construction of families of 0-isospectral lens spaces. For each q and m positive integers, he considered the subfamily of lens spaces (4.8) L0(q; m)={L(q; s1,...,sm):si ≡±sj (mod q) ∀ i = j}. Denote by L0(q; m) the isometry classes in L0(q; m). By the definition, the parameters s1,...,sm of every lens space in an isometry class in L0(q; m)mustbe all different. For L = L(q; s1,...,sm) ∈ L0(q; m), choose h integerss ¯1,...,s¯h such that {±s1,...,±sm, ±s¯1,...,±s¯h} is a set of representatives of integers mod q,coprimetoq. Therefore 2m+2h = φ(q), where φ(q) denotes the Euler phi function. Denote by L¯ the 2h−1-dimensional lens space L(q;¯s1,...,s¯h)andby¯γ the generator of the group Γ¯ given by (4.6) withs ¯i 2h−1 in place of si,thusL¯ = Γ¯\S .ItiseasytoshowthattwolensspacesL and L in L0(q; m) are isometric if and only if L and L are isometric ([Ik88, Prop. 3.3]). Furthermore, Ikeda proved the following important fact. Proposition 4.4. Let q be an odd prime. Two lens spaces L, L ∈ L0(q; m) are p-isospectral for all p if and only if L and L are p-isospectral for all p.

Ikeda restricted his* attention to lens spaces in L0(q; m)forq an odd prime. m − In this case, each term (z − ξsj l)(z − ξ sj l) in (4.7) divides the q-th cyclo- j=1* q−1 − l ≤ ≤ − tomic polynomial Φq(z):= l=1 (z ξ ) for any 1 l q 1. Hence, for 2m−1 L = L(q; s1,...,sm)=Γ\S ∈ L0(q; m), (4.7) implies that − 2 2 q 1 &h 1 1 − z 1 − z − 0 − − s¯j l − s¯j l (4.9) FΓ(z)= 1+ 2m + (z ξ )(z ξ ). q (1 − z) q Φq(z) l=1 j=1 Set q−1 &h s¯j l −s¯j l (4.10) ΨΓ(z)= (z − ξ )(z − ξ ). l=1 j=1 This is a polynomial of degree 2h with coefficients in the cyclotomic field Q(ξ). Now, (4.9) gives a finite condition for 0-isospectrality, namely, L, L ∈ L0(q; m)are 0-isospectral if and only if ΨΓ(z)=ΨΓ (z). In this way, Ikeda found families of 0-isospectral lens spaces by showing that, in some cases, the (well defined) map 2m−1 L =Γ\S ∈ L0(q; m) −→ ΨΓ(z) ∈ Q(ξ)[z] is not one to one (see [Ik80a, Thm. 3.1]). To find such examples, he first compute some coefficients of ΨΓ(z)(see[Ik80a, Prop. 1.2]). Indeed, let q be an odd 2m−1 prime and q − 1=2m +2h.ForL =Γ\S ∈ L0(q; m), if we write ΨΓ(z)=

104 E. A. LAURET, R. J. MIATELLO, AND J. P. ROSSETTI 2h − k 2h−k − − − k=0( 1) ak z , then he shows that a0 = q 1, a1 = 2m, a2 = m(q 2m+1) and ak = a2h−k for all 0 ≤ k ≤ 2h.Notethat,a0, a1, a2, a2h−2, a2h−1 and a2h do not depend on L,thus,ifh =2,ΨΓ(z) is the same for all lens spaces. As a consequence he obtained the following result ([Ik80a, Thm. 3.1]).

Theorem 4.5. Let q be an odd prime and let m be such that 2m +4=q − 1 (i.e. m =(q − 5)/2). Then, any two lens spaces in L0(q, m) are 0-isospectral.

The previous theorem gives a way to obtain increasing families of pairwise 0-isospectral lens spaces of dimension n ≥ 5, where q runs over the odd prime | |≥ 1 (q−1)/2 q−3 numbers. Indeed, L0(q, m) (q−1)/2 m = 4 under the hypotheses in Theorem 4.5. The simplest case is q = 11, thus m = 3 and dimension n =5.Then one can check that L0(q; 3) has two isometry classes represented by L(11; 1, 2, 3) and L(11; 1, 2, 4). One can also check that they are homotopically equivalent to each other by Proposition 4.3. However, if one takes q = 13, then m =4,n =7 and L0(q; 4) contains L(13; 1, 2, 3, 4), L(13; 1, 2, 3, 5) and L(13; 1, 2, 3, 6), which are not homotopically equivalent to each other. Ikeda proved that two non-isometric lens spaces in L0(q; m) as in Theorem 4.5 cannot be p-isospectral for all p (see [Ik88, Thm. 3.9]). The argument is as follows. Suppose that L, L ∈ L0(q; m)arep-isospectral for all p, with q an odd prime and q − 1=2m + 4. By Proposition 4.4, L and L are p-isospectral for all p and of dimension 2h − 1 = 3. However, two 3-dimensional 0-isospectral lens spaces must beisometric(see[IY79], [Ya80]), thus L and L, and therefore L and L,are isometric. However, by using the same family as in Theorem 4.5, Ikeda in [Ik88] found for each p0 ≥ 0, examples of pairs of lens spaces that are p-isospectral for every 0 ≤ p ≤ p0 but are not p0 + 1-isospectral. We conclude this section by giving the main ideas used in his proof. We will use the condition of p-isospectrality for 2m−1 0 ≤ p ≤ p0 in Proposition 4.1. Let L = L(q; s1,...,sm)=Γ\S ∈ L0(q; m), where q is an odd prime number and q −1=2m+4. Similarly as in (4.9) we obtain that

− 2m q 1 p l p p * χ (g ) (4.11) F (z)= + m − Γ − 2m − sj l − sj l (z 1) j=1(z ξ )(z ξ ) l=1 2m = p − Φ (z)−1(z − 1)4 (z − 1)2m q q &2 −1 p l s¯j l −s¯j l +Φq(z) χ (g ) (z − ξ )(z − ξ ). l=1 j=1

Hence, the polynomial

q &2 − p p l − s¯j l − s¯j l ΨΓ(z):= χ (g ) (z ξ )(z ξ ), l=1 j=1

NON-STRONGLY SPHERICAL SPACE FORMS 105 which has degree four, determines Fp(z), thus the ﬁve coeﬃcients of Ψp (z)= Γ Γ 4 − t (t) t t=0( 1) bL,p z play an important role. One can check that (see [Ik88, p. 404]) [p/2] (0) (4) − m (p−2d) bL,p = bL,p = 1+q d AL (0), d=0 [p/2] , / (1) (3) − m (p−2d) (p−2d) bL,p = bL,p = 1+2q d AL (¯s1)+AL (¯s2) , d=0 [p/2] , / (2) − (0) m (p−2d) (p−2d) − bL,p = 1+2bL,p +2q d AL (¯s1 +¯s2)+AL (¯s1 s¯2) , d=0 where ≡− ∀ ∈ (k) ⊂ a a (mod q) a = a A, (4.12) AL (s):=# A S : | | ≡ . A = k, a∈A a s (mod q) and S = {±s1,...,±sm}. Now, Proposition 4.1 can be rewritten in the particular case of lens spaces in L0(q; m) as follows (see [Ik88, Prop. 4.2]).

Proposition 4.6. Let q be an odd prime, m =(q − 5)/2 and 0 ≤ p0 ≤ m − 1. Then, two lens spaces L = L(q; s1,...,sm) and L = L(q; s1,...,sm) in L0(q; m) are p-isospectral for all 0 ≤ p ≤ p if and only if ⎧ 0 ⎪ (p) (p) ⎨ AL (0) = AL (0), (p) (p) (p) (p) (4.13) A (¯s )+A (¯s )=A (¯s )+A (¯s ), ⎩⎪ L 1 L 2 L 1 L 2 (p) (p) − (p) (p) − AL (¯s1 +¯s2)+AL (¯s1 s¯2)=AL (¯s1 +¯s2)+AL (¯s1 s¯2), for all 0 ≤ p ≤ p0.

Ikeda found subfamilies in L0(q; m) such that satisfy (4.13). Set a1s¯1 + a2s¯2 ≡ 0(modq), (4.14) Lp(q; m)= L(q; s) ∈ L0(q; m): , ∀ 1 ≤|a1| + |a2|≤p +2 thus one has the ﬁltration

(4.15) L0(q; m) ⊃ L1(q; m) ⊃ L2(q; m) ⊃ ... .

For example, for q ≥ 11 an odd prime, if L = L(q;1, 2) then L ∈ L0(q; m)L1(q; m) and if L = L(q;1, 3) then L ∈ L1(q; m) L2(q; m). By making several computations with the numbers in (4.12), he showed that two lens spaces at the same level p0 of the ﬁltration satisfy (4.13) for all 0 ≤ p ≤ p0. Moreover, if only one of them lies in the next level p0 + 1, then they cannot satisfy (4.13) for p = p0 +1.Moreprecisely,wecannowstate[Ik88,Thm.4.1].

Theorem 4.7. Let q be an odd prime, m =(q − 5)/2 and 0 ≤ p0 ≤ m − 1 and let L and L be lens spaces in Lp0 (q; m).ThenL and L are p-isospectral for all ≤ ≤ ∈ ∈ 0 p p0.IffurthermoreL Lp0+1(q; m) and L / Lp0+1(q; m),thenL and L are not p0 +1-isospectral.

Now we ﬁx p0 ≥ 0. Let q be a prime number greater than (p0 +2)(p0 +3)+1 and set m =(q − 5)/2. If L and L are the lens space in L0(q; m) such that L = L(q;1,p0 +2), L = L(q;1,p0 +3)∈ L0(q; 2), then one can check that L ∈ ∈ Lp0 (q; m) Lp0+1(q; m)andL Lp0+1(q; m). As a consequence one obtains the following corollary ([Ik88, Thm. 4.10]).

106 E. A. LAURET, R. J. MIATELLO, AND J. P. ROSSETTI

Corollary 4.8. For each p0 ≥ 0 there are lens spaces that are p-isospectral for all 0 ≤ p ≤ p0 but are not p0 +1-isospectral.

5. Isospectral lens spaces and ·1-isospectral lattices In this section we will explain the remarkable relation between isospectrality of lens spaces and isospectrality of integral lattices with respect to the one-norm, introduced in [LMR15b]. Using this connection we were able to find examples of lens spaces p-isospectral for all p which are not coming from Sunada’s method, and are far from being strongly isospectral. Indeed, we present an infinite family of pairs of 5-dimensional lens spaces with these properties, and as a byproduct, an infinite family of such pairs in increasing dimensions. Finally, by means of a finite-implies-infinite principle (see [LMR15b, §4]), we find —with the help of the computer— many more such pairs in low dimensions. We first present the main ideas and the results without proofs, so that the reader can get to them quickly, and after this we develop the mathematical arguments supporting these results. We naturally associate to a lens space L = L(q; s1,s2,...,sm) of dimension 2m − 1 the integral lattice L = L(q; s1,...,sm)ofrankm given by the congruence equation

(5.1) (a1,...,am) ∈L ⇐⇒a1s1 + ···+ amsm ≡ 0(modq). We call a lattice of this kind a congruence lattice. We consider the one norm, i.e. (a1,...,am)1 := |a1| + ···+ |am|. By using Proposition 4.3 it is easy to prove that two lens spaces are isometric if and only if their associated congruence lattices are ·1-isometric (see [LMR15b, Prop. 3.3]). It was surprising to discover that the isospectrality of two lens spaces is directly connected with the isospectrality in one-norm of the associated lattices (see [LMR15b, Thm. 3.9(i)]). Theorem 5.1. Two lens spaces are 0-isospectral if and only if the associated congruence lattices are ·1-isospectral. If one considers one individual p only, p-isospectrality for two lens spaces, does not correspond to a clean and neat condition on the associated lattices, as in the previous theorem for p = 0. However, the condition of being p-isospectral for all p simultaneously turns out to correspond —again as a happy surprise— to a nice ·∗ geometric condition on the associated lattices, which we call 1-isospectrality:for each k and , both lattices must have the same number of vectors with one-norm equal to k and zero coordinates. Theorem 5.2. Two lens spaces are p-isospectral for all p if and only if the ·∗ associated congruence lattices are 1-isospectral. The proofs of these theorems use representation theory of compact Lie groups and properties of the weight lattice. The ideas are given in the next subsection. ·∗ The basic example of 1-isospectral congruence lattices is the pair (5.2) L(49; 1, 6, 15) and L(49; 1, 6, 20). These 3-dimensional lattices produce two 5-dimensional non-isometric lens spaces which in light of the previous theorem are p-isospectral for every p.Thisexample is the ﬁrst one of the following inﬁnite family of pairs (5.3) L(r2t;1,rt− 1, 2rt +1) and L(r2t;1,rt− 1, 3rt − 1),

NON-STRONGLY SPHERICAL SPACE FORMS 107 for r, t ∈ N with r not divisible by 3 (see [LMR15b, Thm. 6.3]). For some purposes, it will be convenient to write this in the following equivalent way

(5.4) L(r2t;1, 1+rt, 1+3rt)andL(r2t;1, 1 − rt, 1 − 3rt),

From these examples, it is possible to construct examples in arbitrarily high dimensions by using Proposition 5.3. In this way, we obtain for each pair of (2m−1)- dimensional lens spaces in our family, another pair of (2h − 1)-dimensional lens spaces with 2h +2m = φ(q), q = r2t, that are again p-isospectral for every p.For example when t =1andr is prime, the dimension increases from 5 to 2h − 1= r2 − r − 7. For the basic pair when r = 7 one has that h =18and2h − 1 = 35. These examples are not the only existing ones, as one can guess. We proved ·∗ in [LMR15b, Thm. 4.2] that to check 1-isospectrality, it suffices to check it in a finite cube, which means that only finitely many computations are enough to ensure p-isospectrality for all p of the lens spaces. By using this, with the help of a computer, we found many more examples. Moreover, one can find all the existing examples for given fixed m and q (see Tables 1, 2 and 3). However, the computing time grows rapidly with m.

5.1. Characterization theorems. In this subsection we give the ideas lead- ing to Theorems 5.1 and 5.2 and sketch their proofs. The approach is based on representation theory of compact Lie groups. From Section 3, G =SO(2( m), K =SO(2m − 1), T is the standard maximal m Z Zm torus of G,andP (G)= j=1 εj is the weight lattice of G. Aswehaveseen \ 2m−1 Γ in Theorem 3.1, the τ-spectrum of Γ S is determined by the numbers dim Vπ for every π ∈ G) such that [τ : π]=1.Anyπ ∈ G) decomposes as a sum of weight spaces under the action of T as Vπ = Vπ(η). η∈P (G)

The multiplicity of a weight η ∈ P (G)inπ is mπ(η):=dimVπ(η). If Γ ⊂ T ,it follows that Γ Γ (5.5) dim Vπ = dim Vπ(η) = mπ(η)

η∈P (G) η∈LΓ

η m where LΓ = {η ∈ P (G):γ =1 ∀ γ ∈ Γ}, which is a sublattice of P (G) Z depending only on Γ but not on π.Hereγη denotes the scalar for which γ acts on Vπ(η). 2m−1 A lens space L(q; s1,...,sm)=Γ\S satisﬁes that Γ ⊂ T since it is gener- ··· 2πi a1s1+ +amsm ∈ η q ∈ ated by γ T as in (4.6). Since γ = e for η = j aj εj P (G), we have that LΓ = L(q; s1,...,sm)deﬁnedin(5.1).

Sketch of proof of Theorem 5.1. One can show that (see, for instance,

[LMR15b, Lemma 3.6]), when π = πkε1 is the irreducible representation of SO(2m) m with highest weight kε1, the multiplicity of η ∈ Z in π is r+m−2 − ∈ N m−2 if η 1 = k 2r with r 0, (5.6) mπ (η)= kε1 0otherwise.

108 E. A. LAURET, R. J. MIATELLO, AND J. P. ROSSETTI

In particular, m (η) depends only on η . From (5.5) and (5.6) it follows that πkε1 1 [k/2] Γ r+m−2 (5.7) dim V = − πkε1 m 2 r=0 η∈L: − η 1=k 2r [k/2] r+m−2 { ∈L − } = m−2 # η Γ : η 1 = k 2r . r=0 Moreover, by Theorem 3.2, this number is exactly the multiplicity of the eigenvalue 2 − \ 2m−1 λk = k + k(2m 2) of the Laplace-Beltrami operator Δτ0,Γ on Γ S .This clearly shows that two lens spaces are 0-isospectral if their associated lattices are ·1-isospectral, thus proving the converse assertion in Theorem 5.1. The remaining implication is proved by induction on k (see [LMR15b, Thm. 3.9(i)]). Sketch of proof of Theorem 5.2. We proceed as in the previous theorem but in this case there are more difficulties. By Corollary 3.3 (iii), we have to show Γ Γ that dim V =dimV for all k and p,whereπk,p is as in Section 3. Now, πk,p πk,p dim V Γ = m (η), but in this case we do not know an explicit formula πk,p η∈L πk,p like (5.6) for mπk,p (η) for arbitrary k and p. Fortunately, this difficulty could be overcome thanks to the following cute regularity property of the multiplicities: two weights with the same one-norm and the same number of zero coordinates have the same multiplicity. This was proved in [LMR15b, Lem. 3.7] with techniques of representation theory of compact Lie groups. Analogously to (5.6), by using the above property we obtain that (see [LMR15b, Thm. 3.8]) [(k+p)/2] m Γ (5.8) dim V = m (μ ) NL(k + p − 2r, ), πk,p πk,p r, r=0 =0 where μr, is any weight in L with μr,1 = k+p−2r and having zero coordinates, and NL(r, ) denotes the number of weights with zero coordinates and one-norm equal to r. Now, clearly, the converse of Theorem 5.2 follows. The other assertion can again be proved by induction on k (see [LMR15b, Thm. 3.9(ii)]). ·∗ 5.2. Construction of 1-isospectral lattices. The characterization of lens spaces p-isospectral for all p given in Theorem 5.2, motivated us to look for exam- ·∗ ples of 1-isospectral congruence lattices. It seems interesting that one can work on the construction of such examples by just working on lattices, without any use of differential geometry. For r, t ∈ N, r>1 not divisible by 3, we set q = r2t and consider the lattices in (5.4), L = L(q;1, 1+rt, 3rt +1)and L = L(q;1, 1 − rt, 1 − 3rt). This is an ·∗ Zm infinite two-parameter family of pairs of 1-isospectral lattices in for m =3 ([LMR15b, Thm. 6.3]). For r ≥ 7 they are not ·1-isometric (see [LMR15b, Lemma 5.4]). We note that the dimension m = 3 of these examples is minimal, since Ikeda and Yamamoto showed that such pairs cannot exist in dimension m =2 ([IY79], [Ya80]). The first step in the proof is to reduce the problem to show that the lattices are just ·1-isospectral, since one can verify that, for 1 ≤ ≤ 3, the number of

NON-STRONGLY SPHERICAL SPACE FORMS 109 elements in L and L with zero coordinates and a fixed one-norm coincide (see [LMR15b, Lemma 6.1]). This implies, for the family of pairs L = L(r2t;1, 1+ rt, 1+3rt),L = L(r2t;1, 1 − rt, 1 − 3rt), the pleasant fact in spectral geometry that: L and L are p-isospectral for all p if and only if L and L are 0-isospectral. According to the previous paragraph, it is sufficient to check that L and L are ·1-isospectral. More precisely, NL(k, )=NL (k, ), where NL(k, ) denotes the number of η ∈Lwith η1 = k and zero coordinates. By a careful calculation of these numbers, we check they coincide. 5.3. Examples in arbitrarily large dimensions. We will show that the infinite family of pairs in dimension 5 given in the previous section allows to produce an infinite family of pairs in arbitrarily large dimensions. For this, we prove an extension of Proposition 4.4 for q = r2, r prime. We recall from (4.8) that L0(q, m) stands for the set of lens spaces of dimension 2m − 1, fundamental group of order q and different parameters. For L =Γ\S2m−1, the function QL(w, z):=QΓ(w, z) given in (4.4) characterizes all p-spectrum. If k ∈ q−1 det(w−γ ) { k L = L(q; s1,...,sm) L0(q, m)thenQL(w, z)= k=0 det(z−γk) since Γ = γ : 0 ≤ k ≤ q − 1}. 2 Proposition 5.3. Let q = r with r prime and let L = L(q; s1,...,sm) and L =(q; s1,...,sm) be lens spaces in L0(q, m) such that sj ≡±1(modr) and ≡± sj 1(modr) for all j.Then,L and L are p-isospectral for all p if and only if L and L are p-isospectral for all p.

Proof. We will obtain in (5.10) a useful relation connecting QL(w, z)and ∈ ≡± QL¯ (w, z)forL = L(q; s1,...,sm) L0(q, m) such that sj 1(modr) for all j. Hereγ ¯ ands ¯1,...,s¯h are as in §4, the paragraph before Prop. 4.4. We have − r 1 det w − (γr)l det(w − γk) (5.9) QL(w, z)= + . − r l det(z − γk) l=0 det z (γ ) gcd(k,r)=1

2 r ±2πirsj /r ±2πi/r The eigenvalues of γ are e = e for 1 ≤ j ≤ m since rj ≡±1 lr r−1 det(w−γ ) r\ 2m−1 (mod r), thus l=0 det(z−γlr) = QL0 (w, z)whereL0 stands for γ S .One can check that L0 is isometric to L(r;1,...,1), hence QL0 (w, z) does not depend on L. k k Since r is prime, det(z − γ )det(z − γ¯ )isequaltoΦq(z)ifgcd(k, q)=1,to r r2−r 2 Φr(z) if gcd(k, q)=r,andto(z − 1) if gcd(k, q)=r . Hence,

k Φq(w) det(z − γ¯ ) (5.10) QL(w, z)=QL0 (w, z)+ k Φq(z) det(w − γ¯ ) gcd(k,r)=1 − Φ (w) (z − 1)2h r 1 det z − (¯γr)l q − − = QL0 (w, z)+ QL(z,w) 2h l Φq(z) (w − 1) det w − (¯γr) ) " "l=1 ## − 2h r − 2m Φq(w) − (z 1) − Φr(z) − (w 1) =QL0 (w, z)+ QL(z,w) 2h r QL0 (w, z) 2m . Φq(z) (w − 1) Φr(w) (z − 1)

Thus, the last expression for QL(w, z)involvesQL¯ (z,w) and other functions which do not depend on L. This clearly shows that QL(w, z)andQL(w, z) determine each other in this case. In particular, QL(w, z)=QL (w, z) if and only if QL(w, z)= QL (w, z), thus the assertion follows from Theorem 4.2.

110 E. A. LAURET, R. J. MIATELLO, AND J. P. ROSSETTI

As a corollary we can now state ([LMR15b, Thm. 7.3])

Theorem 5.4. For any n0 ≥ 5, there exist pairs of non-isometric lens spaces of dimension n,withn ≥ n0, that are p-isospectral for all p. Proof. For each odd prime r ≥ 7sett =1andq = r2. The corresponding 5-dimensional lens spaces L, L ∈ L0(q; 3) from (5.4) are p-isospectral for all p,by Theorem 5.2. By Proposition 5.3, L and L are p-isospectral for all p and have dimension 2h − 1=φ(r2) − 7=r2 − r − 7. This quantity tends to inﬁnity when r does, thus the assertion in the theorem follows.

We recall from [LMR15b, Thm. 7.3] that, by using our 5-dimensional examples, one can construct, in every dimension n ≥ 5, pairs of n-dimensional Riemann- ian manifolds that are p-isospectral for all p and are not strongly isospectral. Remark 5.5. We are interested in the question whether one can extend the duality property in Proposition 5.3 for more general values of q. (We have checked that all the pairs dual to the pairs in the tables remain p-isospectral for all p.) 5.4. Computations and tables. We will show tables with many examples of pairs of lens spaces p-isospectral for all p in low dimensions n =5,7and9. The finite-implies-infinite principle mentioned above allowed us to give an al- gorithm —implemented in Sage [Sa]— that can find, for each m and q, all pairs of lens spaces of dimension n =2m − 1 and fundamental group of order q that are p-isospectral for all p. This is shown in the tables for n =5andq ≤ 500, n =7 and q ≤ 300, and n =9andq ≤ 150. On the other hand, Peter Doyle has implemented a clever computer program using the function QL(w, z) in Theorem 4.2 that can distinguish very quickly whether two lens spaces are p-isospectral for all p. We thank Peter for verifying with his method that all of our examples are correct. For positive integers r and t,andq = r2t (see [LMR15b, §5]) we introduce the element θ := rt +1. Clearly θk ≡ krt +1 (modq), thus θr ≡ 1(modq). Since the parameters in most of the lens spaces occurring in the low dimensional examples are congruent to ±1(modrt), they can be written as powers of θ. In this way, the basicexamplecanbewrittenas − ∼ − − L(49; 1, 6, 15) = L(49; θ0, −θ1,θ 2) = L(q; θ0,θ 1,θ 3), (5.11) ∼ L(49; 1, 6, 20) = L(49; θ0, −θ1, −θ3) = L(q; θ0,θ1,θ3). ∼ Here = means isometry between lens spaces. Each entry q, r, t, (d0,...,dm)inthe tables, represents the pair of lens spaces − − − (5.12) L(q; θd0 ,θd1 ,...,θdm−1 )andL(q; θ d0 ,θ d1 ,...,θ dm−1 ), which are p-isospectral for all p. When an entry has a dag †, it means that the pair corresponding to this line is isometric to the pair of the previous line. This only happens when t is not square-free, thus it is possible to write q as r2t in more than one way. Question 5.1 in [LMR15b] asked for conditions on d =(d0,...,dm−1), r and t, so that the pair of lens spaces in (5.12) are p-isospectral for every p. Peter Doyle worked on this question in collaboration with D. DeFord (see [DD14]) and found sufficient conditions by using in a clever way similar techniques as Ikeda.

NON-STRONGLY SPHERICAL SPACE FORMS 111

Table 1. Pairs of lens spaces p-isospectral for all p of dimension n = 5 and fundamental group of order q ≤ 500.

qrtd0,d1,d2 qrtd0,d1,d2 qrtd0,d1,d2 49 7 1 0, 1, 3 256 16 1 0, 2, 7 361 19 1 0, 2, 7 64 8 1 0, 1, 3 256 16 1 0, 3, 9 361 19 1 0, 2, 8 98 7 2 0, 1, 3 289 17 1 0, 1, 3 361 19 1 0, 2, 9 100 10 1 0, 1, 3 289 17 1 0, 1, 4 361 19 1 0, 2,10 100 10 1 0, 1, 4 289 17 1 0, 1, 5 361 19 1 0, 3, 7 121 11 1 0, 1, 3 289 17 1 0, 1, 6 361 19 1 0, 3, 8 121 11 1 0, 1, 4 289 17 1 0, 1, 7 361 19 1 0, 3, 9 121 11 1 0, 1, 5 289 17 1 0, 1, 8 361 19 1 0, 3,10 121 11 1 0, 2, 5 289 17 1 0, 2, 5 361 19 1 0, 4, 9 121 11 1 0, 2, 6 289 17 1 0, 2, 6 361 19 1 0, 4,10 128 8 2 0, 1, 3 289 17 1 0, 2, 7 361 19 1 0, 4,11 147 7 3 0, 1, 3 289 17 1 0, 2, 8 361 19 1 0, 5,11 169 13 1 0, 1, 3 289 17 1 0, 2, 9 363 11 3 0, 1, 3 169 13 1 0, 1, 4 289 17 1 0, 3, 7 363 11 3 0, 1, 4 169 13 1 0, 1, 5 289 17 1 0, 3, 8 363 11 3 0, 1, 5 169 13 1 0, 1, 6 289 17 1 0, 3, 9 363 11 3 0, 2, 5 169 13 1 0, 2, 5 289 17 1 0, 4, 9 363 11 3 0, 2, 6 169 13 1 0, 2, 6 289 17 1 0, 4,10 384 8 6 0, 1, 3 169 13 1 0, 2, 7 294 7 6 0, 1, 3 392 14 2 0, 1, 3 169 13 1 0, 3, 7 300 10 3 0, 1, 3 392 14 2 0, 1, 4 192 8 3 0, 1, 3 300 10 3 0, 1, 4 392 14 2 0, 1, 5 196 14 1 0, 1, 3 320 8 5 0, 1, 3 392 14 2 0, 1, 6 196 14 1 0, 1, 4 324 18 1 0, 1, 5 392 14 2 0, 2, 5 196 14 1 0, 1, 5 324 18 1 0, 1, 8 392 14 2 0, 2, 6 196 14 1 0, 1, 6 324 18 1 0, 2, 7 39278† 0, 1, 3 196 14 1 0, 2, 5 338 13 2 0, 1, 3 392 14 2 0, 3, 8 196 14 1 0, 2, 6 338 13 2 0, 1, 4 400 20 1 0, 1, 3 196 7 4† 0, 1, 3 338 13 2 0, 1, 5 400 20 1 0, 1, 7 196 14 1 0, 3, 8 338 13 2 0, 1, 6 400 20 1 0, 2, 6 200 10 2 0, 1, 3 338 13 2 0, 2, 5 400 10 4† 0, 1, 3 200 10 2 0, 1, 4 338 13 2 0, 2, 6 400 20 1 0, 2, 8 242 11 2 0, 1, 3 338 13 2 0, 2, 7 400 10 4† 0, 1, 4 242 11 2 0, 1, 4 338 13 2 0, 3, 7 400 20 1 0, 2, 9 242 11 2 0, 1, 5 343 7 7 0, 1, 3 400 20 1 0, 3, 9 242 11 2 0, 2, 5 361 19 1 0, 1, 3 441 21 1 0, 1, 5 242 11 2 0, 2, 6 361 19 1 0, 1, 4 441 21 1 0, 2,10 245 7 5 0, 1, 3 361 19 1 0, 1, 5 441 21 1 0, 3, 9 256 16 1 0, 1, 3 361 19 1 0, 1, 6 44179† 0, 1, 3 256 16 1 0, 1, 6 361 19 1 0, 1, 7 448 8 7 0, 1, 3 256 16 1 0, 1, 7 361 19 1 0, 1, 8 484 22 1 0, 1, 3 256 16 1 0, 2, 5 361 19 1 0, 1, 9 484 22 1 0, 1, 4 256 16 1 0, 2, 6 361 19 1 0, 2, 5 484 22 1 0, 1, 5 256 8 4† 0, 1, 3 361 19 1 0, 2, 6 484 22 1 0, 1, 6

continued

112 E. A. LAURET, R. J. MIATELLO, AND J. P. ROSSETTI

qrtd0,d1,d2 qrtd0,d1,d2 qrtd0,d1,d2 484 22 1 0, 1, 7 484 22 1 0, 2, 9 484 11 4† 0, 2, 5 484 22 1 0, 1, 8 484 22 1 0, 2,10 484 22 1 0, 4,12 484 22 1 0, 1, 9 484 11 4† 0, 1, 5 484 11 4† 0, 2, 6 484 22 1 0, 1,10 484 22 1 0, 3, 7 484 22 1 0, 5,12 484 22 1 0, 2, 5 484 22 1 0, 3, 8 484 22 1 0, 5,13 484 22 1 0, 2, 6 484 22 1 0, 3, 9 484 22 1 0, 6,13 484 11 4† 0, 1, 3 484 22 1 0, 3,10 490 7 10 0, 1, 3 484 22 1 0, 2, 7 484 22 1 0, 3,12 500 10 5 0, 1, 3 484 22 1 0, 2, 8 484 22 1 0, 4, 9 500 10 5 0, 1, 4 484 11 4† 0, 1, 4 484 22 1 0, 4,10

To state their condition, in their terminology, d =(d0,...,dm−1)issaidtobe: • univalent mod r if its entries are distinct mod r, • reversible mod r if the pair of lens spaces in (5.12) are isometric, • good mod r if it is univalent or reversible mod r, • hereditarily good mod r if it is good mod c for all c dividing r. Then, Theorem 1 in [DD14]) says that if d is hereditarily good mod r and not reversible mod r,then the lens spaces in (5.12) are p-isospectral for all p and are not isometric. By using this condition it is remarkably simple to produce examples. Indeed, the family in (5.4) satisﬁes the conditions of this theorem. The pairs in (5.4) has d =(0, 1, 3) from (5.11). Now, d is clearly univalent mod r for r ≥ 4, reversible mod r for r =1, 2, 4, 5, thus d is good mod r for any r =3.Consequently, d is hereditarily good mod r for any r not divisible by 3, and not reversible for r ≥ 7. We believe that this theorem is an important step in the determination of all the lens spaces that are p-isospectral for all p. Though many such examples do come from the theorem, there are exceptions like the ones we next describe.

d0 d1 dm−1 Example 5.6. To each lens space L(q; θ ,θ ,...,θ ) with 0 = d0

r =(d1 − d0)+···+(dm−1 − dm−2)+(r − dm−1). One can check that two lens spaces are isometric if their partitions diﬀer by a cyclic reordering. Not all the known examples of p-isospectral lens spaces for all p can be written as in (5.12). For instance, this is the case for the pair L, L, dual to the basic pair (5.2), since their parameters are not necessarily congruent to ±1(modrt). Moreover, this phenomenon already occurs in the case of the curious example L(72; 1, 5, 7, 17, 35), L(72; 1, 5, 7, 19, 35), since neither these lens spaces nor their du- als (namely L(72; 1, 5, 7, 11, 19, 25, 35) and L(72; 1, 5, 7, 11, 23, 29, 31)) can be written as in (5.12). Problem 5.7. Determine all pairs of n-dimensional lens spaces p-isospectral for all p with fundamental group of order q (or at least all such pairs for inﬁnite values of n). Question 5.8. Are there families of non-isometric lens spaces p-isospectral for all p having more than two elements?

NON-STRONGLY SPHERICAL SPACE FORMS 113

Table 2. Pairs of lens spaces p-isospectral for all p of dimension n = 7 and fundamental group of order q ≤ 300.

qrtd0,d1,d2,d3 qrtd0,d1,d2,d3 qrtd0,d1,d2,d3 49 7 1 0, 1, 2, 4 169131 0,3,4,8 256 16 1 0, 1, 2, 7 81 9 1 0, 1, 2, 4 196141 0,1,2,4 256 16 1 0, 1, 3, 6 81 9 1 0, 1, 2, 5 196141 0,1,2,5 256 16 1 0, 1, 4, 7 81 9 1 0, 1, 3, 5 196141 0,1,2,6 256 16 1 0, 1, 5,10 98 7 2 0, 1, 2, 4 196141 0,1,3,5 256 16 1 0, 2, 3, 7 100101 0,1,2,4 196141 0,1,3,6 256 16 1 0, 2, 3, 9 100101 0,2,3,6 196141 0,1,4,6 256 16 1 0, 2, 5, 9 121111 0,1,2,4 196141 0,1,4,9 256 16 1 0, 3, 4, 9 121111 0,1,2,5 196141 0,1,5,9 256 16 1 0, 3, 5,10 121111 0,1,2,6 196141 0,2,3,6 288 12 2 0, 1, 2, 5 121111 0,1,3,5 196141 0,2,3,8 288 12 2 0, 2, 3, 7 121111 0,1,3,6 196141 0,2,4,8 289 17 1 0, 1, 2, 4 121111 0,1,3,7 19674† 0, 1, 2, 4 289 17 1 0, 1, 2, 5 121111 0,1,4,6 196141 0,2,5,8 289 17 1 0, 1, 2, 6 121111 0,1,4,7 196141 0,3,4,8 289 17 1 0, 1, 2, 7 121111 0,2,3,6 196141 0,3,5,9 289 17 1 0, 1, 2, 8 121111 0,2,4,7 200102 0,1,2,4 289 17 1 0, 1, 2, 9 144121 0,1,2,5 200102 0,2,3,6 289 17 1 0, 1, 3, 5 144121 0,2,3,7 225151 0,1,2,4 289 17 1 0, 1, 3, 6 147 7 3 0, 1, 2, 4 225151 0,1,2,6 289 17 1 0, 1, 3, 7 162 9 2 0, 1, 2, 4 225151 0,1,2,8 289 17 1 0, 1, 3, 8 162 9 2 0, 1, 2, 5 225151 0,1,3,7 289 17 1 0, 1, 3, 9 162 9 2 0, 1, 3, 5 225151 0,1,3,8 289 17 1 0, 1, 3,10 169131 0,1,2,4 225151 0,1,4,8 289 17 1 0, 1, 4, 6 169131 0,1,2,5 225151 0,1,5,9 289 17 1 0, 1, 4, 7 169131 0,1,2,6 225151 0,2,4,7 289 17 1 0, 1, 4, 8 169131 0,1,2,7 225151 0,2,4,8 289 17 1 0, 1, 4, 9 169131 0,1,3,5 242112 0,1,2,4 289 17 1 0, 1, 4,10 169131 0,1,3,6 242112 0,1,2,5 289 17 1 0, 1, 5, 7 169131 0,1,3,7 242112 0,1,2,6 289 17 1 0, 1, 5, 8 169131 0,1,4,6 242112 0,1,3,5 289 17 1 0, 1, 5, 9 169131 0,1,4,7 242112 0,1,3,6 289 17 1 0, 1, 5,10 169131 0,1,4,8 242112 0,1,3,7 289 17 1 0, 1, 5,11 169131 0,1,5,7 242112 0,1,4,6 289 17 1 0, 1, 6, 8 169131 0,1,5,8 242112 0,1,4,7 289 17 1 0, 1, 6, 9 169131 0,2,3,6 242112 0,2,3,6 289 17 1 0, 1, 6,10 169131 0,2,3,7 242112 0,2,4,7 289 17 1 0, 1, 6,11 169131 0,2,3,8 243 9 3 0, 1, 2, 4 289 17 1 0, 1, 7, 9 169131 0,2,4,7 243 9 3 0, 1, 2, 5 289 17 1 0, 1, 7,10 169131 0,2,4,8 243 9 3 0, 1, 3, 5 289 17 1 0, 2, 3, 6 169131 0,2,5,9 245 7 5 0, 1, 2, 4 289 17 1 0, 2, 3, 7 169131 0,2,5,8 256161 0,1,2,5 289 17 1 0, 2, 3, 8

continued

114 E. A. LAURET, R. J. MIATELLO, AND J. P. ROSSETTI

qrtd0,d1,d2,d3 qrtd0,d1,d2,d3 qrtd0,d1,d2,d3 289 17 1 0, 2, 3, 9 289 17 1 0, 2, 6,10 289 17 1 0, 3, 6,11 289 17 1 0, 2, 4, 7 289 17 1 0, 2, 6,11 289 17 1 0, 3, 7,11 289 17 1 0, 2, 4, 8 289 17 1 0, 2, 7,10 289 17 1 0, 3, 7,12 289 17 1 0, 2, 4, 9 289 17 1 0, 2, 7,11 289 17 1 0, 4, 5,10 289 17 1 0, 2, 4,10 289 17 1 0, 3, 4, 8 289 17 1 0, 4, 6,11 289 17 1 0, 2, 5, 8 289 17 1 0, 3, 4, 9 294 7 6 0, 1, 2, 4 289 17 1 0, 2, 5, 9 289 17 1 0, 3, 4,10 300 10 3 0, 1, 2, 4 289 17 1 0, 2, 5,10 289 17 1 0, 3, 5, 9 300 10 3 0, 2, 3, 6 289 17 1 0, 2, 5,11 289 17 1 0, 3, 5,10 289 17 1 0, 2, 6, 9 289 17 1 0, 3, 6,10

Table 3. Pairs of lens spaces p-isospectral for all p of dimension n = 9 and fundamental group of order q ≤ 150.

qrtd0,d1,d2,d3,d4 qrtd0,d1,d2,d3,d4 64 8 1 0, 1, 2, 3, 5 121111 0,1,3,4,7 72 ? ? 121111 0,1,3,5,7 81 9 1 0, 1, 2, 3, 5 121111 0,1,3,5,8 81 9 1 0, 1, 2, 4, 5 121111 0,1,3,6,7 81 9 1 0, 1, 2, 4, 6 121111 0,1,3,6,8 121111 0,1,2,3,5 121111 0,1,4,5,7 121111 0,1,2,3,6 121111 0,2,3,5,7 121111 0,1,2,4,5 121111 0,2,3,4,7 121111 0,1,2,4,6 128 8 2 0, 1, 2, 3, 5 121111 0,1,2,4,7 144121 0,1,2,3,5 121111 0,1,2,5,6 144121 0,1,2,4,7 121111 0,1,2,5,7 144121 0,2,3,5,7 121111 0,1,3,4,6

5.5. Final remarks. We end this paper with the following comments. Remark 5.9. It is shown in [LMR15b, Lemma 7.6] that the lens spaces in the family constructed above are homotopically equivalent to each other. However, they cannot be simply homotopically equivalent (see [Co70, §31]) since in this case they would be isometric. Remark 5.10. Despite being p-isospectral for every p, the 5-dimensional lens spaces L and L in our family are ‘very far’ from being strongly isospectral. Strongly isospectral spherical space forms are necessarily τ-isospectral for every representation τ of SO(5). However, in the case at hand, one can explicitly show many representations τ of SO(5) such that lens spaces L, L are not τ-isospectral. If we look at the basic case L = L(49; 1, 6, 15) and L = L(49; 1, 6, 20), in [LMR15b, §8] we show that if π0 is the unitary irreducible representation of SO(6) with highest weight Λ0 =4ε1 +3ε2, then the lens spaces L and L are not τ- isospectral for every irreducible representation τ of SO(5) with highest weight of the form b1ε1 + b2ε2 for 4 ≥ b1 ≥ 3 ≥ b2 ≥ 0.

NON-STRONGLY SPHERICAL SPACE FORMS 115

By computer methods, by using Sage [Sa], we checked that there are many choices π0 with this property, thus providing many other K-types τ such that the lens spaces L and L are not τ-isospectral. In this connection, in [LMR15b]we make the following conjecture: There are only finitely many irreducible representations τ of K = SO(5) such that L and L are τ-isospectral. Remark 5.11. Recently, Sebastian Boldt and the first named author in [BL14] extended the methods in this paper to the Dirac operator on lens spaces admitting a spin structure. As it can be expected, the Dirac case involves more technical difficulties. In this case, one associates to a lens space L = L(q; s1,...,sm) with a fixed spin structure, an affine congruence lattice L.Whenq and m are odd, L admits exactly one spin structure and the associated L is given by L { ∈ 1 Z m ··· ≡ } (5.13) = (a1,...,am) ( 2 + ) :2(a1s1 + + amsm) 0(modq) . The case q even is a bit more involved since L admits two spin structures. In analogy with Theorem 5.1, the authors show that two lens spaces are Dirac isospectral if and only if their associated affine congruence lattices are ·1-isospectral. Furthermore, they are able to construct the following examples: • an increasing family of lens spaces mutually Dirac isospectral with increasing dimension; • an infinite sequence of 7-dimensional lens spaces, each of them with two Dirac isospectral spin structures; • an infinite sequence of pairs of non-isometric 7-dimensional lens spaces admitting exactly one spin structure that are Dirac isospectral. Remark 5.12. All examples in the literature of pairs of isospectral spherical space forms with non-cyclic fundamental group (i.e. lens spaces are not allowed) are obtained by Sunada’s method, hence they are strongly isospectral and correspond to almost conjugate subgroups of SO(2m)([Ik83], [Gi85], [Wo01]). Question 5.13. Can one construct 0-isospectral spherical space forms with non-cyclic fundamental groups that are not strongly isospectral? Note that two 3-dimensional isospectral spherical space forms are isometric ([IY79], [Ya80], [Ik80a]), so the answer is negative in dimension 3. Furthermore, Wolf in [Wo01, Cor. 7.3] showed the non-existence of such examples for any dimension 2m − 1 with m prime (see also [Ik80c, Thm. 3.1 and Thm. 3.9]).

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CIEM–FaMAF, Universidad Nacional de Cordoba,´ 5000-Cordoba,´ Argentina E-mail address: [email protected] CIEM–FaMAF, Universidad Nacional de Cordoba,´ 5000-Cordoba,´ Argentina E-mail address: [email protected] CIEM–FaMAF, Universidad Nacional de Cordoba,´ 5000-Cordoba,´ Argentina E-mail address: [email protected]

Contemporary Mathematics Volume 656, 2016 http://dx.doi.org/10.1090/conm/656/13079

Entrance laws for positive self-similar Markov processes

V´ıctor Rivero

Abstract. In this paper we propose an alternative construction of the self- similar entrance laws for positive self-similar Markov processes. The study of entrance laws has been carried out in previous papers using different techniques, depending on whether the process hits zero in a finite time almost surely or not. The technique here used allows to obtain the entrance laws in a unified way. Besides, we show that in the case where the process hits zero in a finite time, if there exists a self-similar entrance law, then there are infinitely many, but they can all be embedded into a single one. We propose a pathwise extension of this embedding for self-similar Markov processes. We apply the same technique to construct entrance law for other types self-similar processes.

1. Introduction and main result

Let P =(Px,x ≥ 0) be a family of probability measures on Skorohod’s space D+, the space of càdlàg paths defined on [0, ∞[ with values in R+.ThespaceD+ is endowed with the Skohorod topology and D is its Borel σ-field. We will denote by X the canonical process of the coordinates and (Gt,t≥ 0) will be the completed natural filtration generated by X. Assume that under P the canonical process X is a positive self-similar Markov process (pssMp), that is to say that (X, P)isa [0, ∞[-valued strong Markov process and that it has the scaling property: there exists an α>0 such that for every c>0, Law ({cXtc−α ,t≥ 0}, Px) =({Xt,t≥ 0}, Pcx) ∀x ≥ 0. In this case we will say that X is an 1/α-positive self-similar Markov process (α- pssMp). We will assume furthermore that (X, P) is a pssMp for which 0 is a cemetery state. The hitting time of zero will be denoted by T0 =inf{t>0:Xt = 0}.So,thelawP0 will be understood as the law of the degenerated path equal to 0. The importance of self-similar Markov processes resides in the fact, shown by Lamperti [19], that it is the totality of Markov processes that can arise as scaling limits of stochastic processes. Said otherwise, this the class of possible limit Markov processes that can occur upon subjecting a fixed stochastic process to infinite contractions of its space and time scales. Further information about this class of processes and its applications can be found in the review paper [24]and

2010 Mathematics Subject Classiﬁcation. Primary 60G18, 60.62, 60G51. Key words and phrases. Self-similar Markov processes, L´evy processes, entrance laws, recurrent extensions.

c 2016 American Mathematical Society 119

120 VÍCTOR RIVERO chapter 13 in [18]. One particular feature of pssMp is that they are in bijection with real-valued Lévy processes. This useful bijection, that we will next explain, is given through the so-called Lamperti’s transformation, in honour to the celebrated work of Lamperti [20]. A R ∪{−∞}-valued Lévy process is an stochastic process whose paths are càdlàg, the state {−∞} is an absorbing point, and it has stationary and independent increments. The state {−∞} is understood as an isolated point and hence the process hits this state and dies at an independent exponential time ζ, with some parameter q ≥ 0, the case q = 0 is included to allow this time to be infinite a.s. The law of ξ is characterized completely by its Lévy-Khintchine exponent Ψ, which takes the following form (1) σ2 ∞ zξ1 2 zy log E e , 1 <ζ =Ψ(z)=−q + bz + z + e − 1 − zyI{|y|<1} Π(dy), 2 −∞ for any z ∈ iR, where σ, b ∈ R and Π is a Lévy measure satisfying the condition 2 ∧ ∞ R(y 1)Π(dy) < . For background about Lévy processes see [1], [18], [31]. In order to state our main results we recall first a few facts about self-similar Markov processes, Lévy processes and exponential functionals of Lévy processes. It is well known (see Lamperti [20]) that for any 1/α-pssMp, X =(Xt,t ≥ 0), there exists a R ∪{−∞} valued Lévy process ξ independent of the starting point X0, such that , /

(2) Xt1{t0: exp(αξu)du>t ,t≥ 0, 0 with the usual convention inf{∅} = ∞. Lamperti proved that, for any x>0, T0 is finite Px-a.s. if and only if either ζ<∞ a.s., or ζ = ∞ a.s. and limt→∞ ξt = −∞ a.s. Conversely, Lamperti showed that given a Lévy process ξ, the transformation just described gives rise to a 1/α-pssMp. We will refer to this transformation as Lamperti’s transformation. Throughout this paper we will assume P is the reference measure, and under P, X will be a pssMp and ξ the Lévy process associated to it via Lamperti’s transformation. The measures (Px,x > 0) are the a conditional regular version of the law of X given X0 = x. Notice that under Px,ξstarts from log x. This implies that the law of ξ, under Px, is that of ξ +logx under P . Besides, P it follows from Lamperti’s transformation that under x the first hitting time of 0 α ζ P for X, T0, has the same law as x 0 exp(αξs)ds, under . The random variable I, defined by ζ (3) I := exp(αξs)ds, 0 is usually named exponential functional of the Lévy process ξ. Lamperti’s above mentioned result implies that I is a.s. finite if and only if ζ<∞ a.s. or ζ = ∞ and limt→∞ ξt = −∞, a.s. Motivated by defining a pssMp issued from 0, when constructed using Lam- perti’s transformation, there have been several papers studying the existence of what we call here self-similar entrance laws, see for instance [3], [4], [6],[8]and[24] where an account on this topic is provided. This is the object of main interest in

ENTRANCE LAWS FOR POSITIVE SELF-SIMILAR MARKOV PROCESSES 121 this paper. We will say that a family of sigma-ﬁnite measures on (0, ∞), {ηt,t>0}, { X ≥ } is a self-similar entrance law for the semigroup Pt ,t 0 of X if the following are satisﬁed (EL-i) the identity between measures X ηsPt = ηt+s, that is

ηs(dx) Ex [f(Xt),t0,t≥ 0; (EL-ii) there exists an index γ ≥ 0 such that for all s>0, −γ/α ηsf = s η1Hs1/α f,

where f denotes any positive and measurable function, and for c>0,Hc denotes the dilation operator Hcf(x)=f(cx).

In that case, we say that {ηs,s>0}, is a γ-self-similar entrance law, γ-ssel for short, associated to X. Observe that the condition (EL-ii) is equivalent to the apparently more general condition: there is a γ ≥ 0 such that for any c>0,s>0, −γ ηsf = c ηsc−α Hcf for any positive and measurable function f. In this paper our main concern is to describe the family of σ-finite ssel for a pssMp that either hits zero in a finite time or never hits zero and the underlying Lévy process in Lamperti’s transformation drifts towards +∞. In several instances we will assume that there exists a θ ≥ 0 such that

E(exp{γξ1}, 1 <ζ) ≤ 1. Which is equivalent to ask that

E(exp{γξt},t<ζ) ≤ 1, ∀t ≥ 0. Under this condition we will denote by P(θ) the unique probability measure on D such that (θ) θξt P = e P on Ft, for all t ≥ 0. (θ) For θ =0, we will write P instead of P(0). As usual, we will denote by P) the law of the dual L´evy process ξ) = −ξ under P(θ) . We have the following theorem whose proof was partially inspired by Fitzsim- mons’ [11] constructions of excursions measures for pssMp.

Theorem 1. Let ((Xt)t≥0, (Px)x>0) be a 1/α-positive self-similar Markov process and ξ the Lévy process associated to it via Lamperti’s transformation. Assume that X hits zero in a finite time a.s. For γ>0 fixed, the following are equivalent

(i) E(exp{γξ1}, 1 <ζ) ≤ 1; γ (ii) the family of measures (μs ,s>0), deﬁned by " ", / # # 1/α γ −γ/α)(γ) s γ −1 μ f := s E f I α , s I + + for f : R → R measurable, forms a γ-ssel for ((Xt)t≥0, (Px)x>0) , and θ ∞ μ11 < ;

122 V´ICTOR RIVERO

γ P γ (iii) there exists a γ-ssel (ηt ,t > 0) for ((Xt)t≥0, ( x)x>0) , such that η1 is a probability measure. In this case, the measures in (ii) form the unique, up to multiplicative constants, finite γ-ssel for X. Furthermore, when one of the above conditions is satisfied, it is then satisfied for every 0 <β<γ.For any pair (β,β), such that 0 <β <β≤ γ, there exists a constant 0 0,s>0, s β ,β β , s z∈(0,1) α α with , / , / β Γ − α β −1 (β β ) −1 P B − ∈ , / , / α − α β (β β ) dy = y (1 y) 1{0

In the case where E(exp{γξ1}, 1 <ζ)=1, the identity in (ii) has been obtained in [25,26]. The second assertion in the latter Theorem implies that although there are inﬁnitely many ssel, all can be embedded into a single one, namely that with largest self-similarity index. In Section 3, we will see that this can be extended to the level of stochastic processes. Besides, the second part of the Theorem has been observed in the work [14] using completely diﬀerent techniques. An easy extension of the results in the paper [34] shows that any ssel η =(ηt,t > 0) is such that either limt→0 ηt1{(a,∞)} =0, for all a>0, or limt→0 ηt1{(a,∞)} > 0, for all a>0. And the latter holds if and only if there is a γ>0 such that dx (5) η (dy)= P (X ∈ dy, t < T ),y>0,t>0; t 1+γ/α x t 0 (0,∞) x in which case η is a γ-ssel. The above facts and the Proposition 1 in [25]imply the following Corollary, where a more tractable expression for the above ssel is provided.

Corollary 1. Let ((Xt)t≥0, (Px)x>0) be a 1/α-positive self-similar Markov process and ξ the L´evy process associated to it via Lamperti’s transformation. As- sume that X hits zero in a ﬁnite time a.s. For γ>0, such that E(exp{γξ1}, 1 < ζ) < 1, we have that there is a constant cγ ∈ (0, ∞) such that " ", / # # 1/α dx −γ/α)(γ) s γ −1 c E (f(X ),s

Theorem 2. Let ((Xt)t≥0, (Px)x>0) be a 1/α-positive self-similar Markov process and ξ the L´evy process associated to it via Lamperti’s transformation. Assume that X never hits zero and ξ drifts towards ∞. The family of measures (μt,t > 0) deﬁned by the relation " ", / # # s 1/α μ f := E) f I−1 , s I + + for f : R → R measurable, forms a 0-ssel for (Xt,t≥ 0) .

ENTRANCE LAWS FOR POSITIVE SELF-SIMILAR MARKOV PROCESSES 123

In the case where the process ξ has ﬁnite mean 0

2. Proof of Theorems 1 and 2 The proof that (i) implies (ii) in Theorem 1 and the claim in Theorem 2 will be given in a uniﬁed way. We will assume either of the following conditions on the underlying L´evy process ξ (TH1) there exists a θ>0 such that

θξ1 E(e 1{1<ζ}) ≤ 1. (TH2) ξ has an inﬁnite lifetime and drifts towards ∞. It is a standard fact that (TH1) is equivalent to require that

θξt E(e 1{t<ζ}) ≤ 1, ∀t ≥ 0, see e.g. [31] Theorem 25.17. We will say that Cram´er’s condition is satisﬁed with index θ>0 if the equality holds

θξt E(e 1{t<ζ})=1, ∀t ≥ 0. When (TH1) holds, we will denote by P(θ) the unique probability measure such that (θ) θξt P = e P on Ft, for all t ≥ 0. It is easily verified that under P(θ) the canonical process still is a R ∪{∞}-valued Lévy process, see e.g. [31] Chapter 33. Furthermore, under P(θ) the lifetime is infinite a.s. if and only if Cramér’s condition is satisfied. Indeed, we have the equality

(θ) θξt P (t<ζ)=E(e 1t<ζ ) ≤ 1, for all t ≥ 0, and thus if Cramér’s condition is satisfied then, ζ = ∞, P(θ)–a.s. We also have that if Cramér’s condition is satisfied then, under P(θ),ξdrifts towards ∞, which λξ1 in turn follows from the convexity of the mapping λ → log E(e 1{1<ζ}), on the βξ1 set C = {β ∈ R : E(e 1{1<ζ}) < ∞}, see e.g. [31] Chapter 25.

124 V´ICTOR RIVERO

Now, we observe that when the condition (TH2) holds, the process (ξ,P) bears the same properties as (ξ,P(θ)) does when Cramér’s condition is satisfied with an index θ>0. Also, in this setting, Cramér’s condition is trivially satisfied taking θ =0. This simple remark is the unifying point of the proofs of Theorems 1 and 2. In order to give a unified argument, we will say that θ =0, whenever the conditions in (TH2) hold. In that case, Cramér’s condition will be necessarily satisfied, and the respective measure P(0) will be P itself, so no difference will be made. Naturally, the case θ>0 will be exclusive to the setting (TH1). ) )(θ) We will denote by P, and P , respectively, the law of the dual process (−ξt,t≥ (θ) 0) under P, and P(θ) respectively. Observe that the processes (ξ,P)and(ξ,P) ) are in weak duality with respect to the measure Λθ(dx):=e−θxdx, x ∈ R .

Indeed, let f,g : R → R+ measurable functions. Using Fubin’s theorem and a change of variables we get: " # −θx −θx dxe f(x) Ex(g(ξt)1{t<ζ})=E dxe f(x)g(x + ξt)1{t<ζ} R "R #

−θy θξt = E dye f(y − ξt)e g(y)1{t<ζ} R −θy θξt = dye g(y) E f(y − ξt)e 1{t<ζ} R −θy (θ) = dye g(y) E f(y − ξt)1{t<ζ} R −θy E)(θ) = dye g(y) y f(ξt)1{t<ζ} . R

(θ) It follows that the measure Λθ is excessive for both (ξ,P)and(ξ,P) ). Moreover, by taking f ≡ 1 in the above identity, it is easily seen that Cramér’s condition is satisfied for (ξ,P), with an index θ ≥ 0, if and only if the measure is actually (θ) invariant for (ξ,P). Whilst the measure Λθ is invariant for (ξ,P) ) if and only if the lifetime of (ξ,P) is infinite, as can also be seen from the above identity by taking g ≡ 1. We denote by (Y,Qθ) the Kusnetzov process associated to ξ and Λθ,seefor instance [9] Chapter XIX or [32]. Qθ is the unique sigma finite measure on D(R, R), such that Qθ ∈ ∈ ∈ (Yt1 dx1,Yt2 dx2,...,Ytn dxn) θ ··· =Λ (dx1)Qt2−t1 (x1,dx2) Qtn−tn−1 (xn−1,dxn), for all −∞

ENTRANCE LAWS FOR POSITIVE SELF-SIMILAR MARKOV PROCESSES 125 following identities that prove our claim. Qθ (f (Y + x)f (Y + x) ···f (Y + x)) 1 t1 2 t2 n tn − θy1 ··· = dy1e f1(y1 + x) E (f2(ξt2−t1 + y1 + x) f(ξtn−t1 + y1 + x)) R θx −θz ··· = e dze f1(z) E (f2(ξt2−t1 + z) f(ξtn−t1 + z)) R θxQθ ··· = e (f1(Yt1 )f2(Yt2 ) fn(Ytn )) Moreover, the image under time reversal of (Y,Qθ), at any finite time, gives a (θ) process with the same semigroup as (ξ,P) ), see [9] Ch. XIX- no.14. We denote by α and β the birth and death times of (Y,Qθ). Observe that if Cramér’s condition is satisfied with an index θ ≥ 0, then α = −∞, Qθ-a.s. While if (ξ,P) has an infinite lifetime then β = ∞, Qθ-a.s. These are well known facts that come from the above observation that in these cases Λθ is invariant for (ξ,P), (θ) or (ξ,P) ), respectively, see e.g. Theorem 6.7 in [12]. We define (ρ(t),t≥ 0), by t ρ(t)= eαYs ds, t ∈ R . α Under Qθ, the process ρ is almost surely finite. This is obtained by conditioning on the future and applying Proposition 4.7 in [22], to get ζ (θ) Qθ ∞ θ P) αξs ∞ (ρ(t)= , α

(θ) where the third identity is a consequence of the fact that the process (ξ,P) ) either ζ (θ) −∞ αξs ∞ P) drifts towards or has a finite lifetime, and thus I = 0 e ds < , x -a.s. for x ∈ R . Whenever θ = 0, we have by the simple Markov property under Qθ that ∞ ∞ ∞ ∞ αYs ρ( )= , a.s. Indeed, given that ρ(0) < its suffices to prove that 0 e ds is a.s. infinite under Qθ. For, we observe that when θ =0, " ∞ # " ∞ # θ αYs θ αYs Q e ds < ∞, α< 0 < β = Λ (dx) Px e ds < ∞ =0; 0 R 0 where the last identity follows from the fact that under Px the Lévy process ξ drifts to ∞. Let Ct be the time change induced by ρ, that is

Ct =inf{s>0:ρ(s) >t},t≥ 0. θ It follows from the previous discussion that Ct < ∞ Q -a.s. By the theory of time changes developed by Kaspi [17] it follows that the family of measures θ Qθ YCt ηt f := f e , 0 0, is an entrance law for the pssMp X. It is important to mention that for each θ t>0,ηt has the following scaling property. If Hc denotes the dilation operator Hcf(x)=f(cx),x ∈ R, we have the equality for any f : R → R measurable and positive

θ θx (6) ηt Hex f = e ηtexα f.

126 V´ICTOR RIVERO

This fact is an easy consequence of the eﬀect of translations under Qθ, that we mentioned above, as the following calculations show , , / / θ Qθ (x+Y )Ct ηt Hex f = f e , 0 0: exp{α(x + Yu)}du > te } = Ct = Ct(Y ),x∈ R . α The latter fact has as a particular consequence that

θ −θ/α θ (8) ηt f = t η1Ht1/α f, t > 0. θ We will now prove that for t>0 the above constructed measures (ηt ,t > 0) and θ the measures (μt ,t>0) as deﬁned in Theorem 1-(ii) and Theorem 2 are equal. Let q>0. Applying Fubini’s theorem and inverting the time change C, we obtain that for every function f positive and measurable ∞ " ∞ # − − qt θ Qθ qt YCt dte ηt f = dte f(e )1{0

ENTRANCE LAWS FOR POSITIVE SELF-SIMILAR MARKOV PROCESSES 127

Putting together the latter and former identities we get the equality of measures dt dt ηθ(t1/αdz)= μθ(t1/αdz), tθ/α 1 t 1 (θ) with μ1 given by " # 1/α (θ) 1 μθ(dz)=P) ∈ dz zα−θ,z>0. 1 I We deduce therefrom the equality of measures θ θ η1 (dz)=μ1(dz). The claim follows from (8). We should now justify that when θ>0,

θ θ − θ E) α 1 ∞ μ11= (I ) < , but this is a consequence of the Lemma 2 in [26] and Lemma 3 in [27], because these results ensure that this condition is implied by the condition θ ) θξ1 E e 1{1<ζ} = E 1{1<ζ} ≤ 1. This finishes the proof of the implication (i) ⇒ (ii) in Theorem 1 and the claim in Theorem 2. 2.1. Continuation of the proof of Theorem 1. That (ii) implies (iii) is straightforward. We are left to prove that (iii) implies (i). In the paper [13], Lemma 5.2, it has been proved that there exists a bijection between the family of γ–ssel for a pssMp X, with γ>0, for which the measure corresponding to the time index 1 is a probability measure, and the family of quasi- stationary laws for the Orstein-Uhlenbeck type process −αt ≤ U U := (Ut = e Xet−1, 0 T0 := log(1 + T0)). Recall that a probability measure ν is a quasi-stationary law for U if we have the equality of measures ν(dx) P (U ∈ dy, t < T ) (0,∞) x t 0 P = ν(dy). ∞ ν(dx) x(t0 such that (0,∞) ν(dx) x(t0 and −γt ν(dx) Px(Xt ∈ dy, t < T0)=e ν(dy). (0,∞) The mentioned bijection is as follows. Given ν a quasi-stationary law for U, as above, the family of measures defined by γ −γ/α ηs f := s νHs1/α f,s > 0. constitutes a γ-ssel for X such that η11=1. Reciprocally, given a γ-ssel for X,such that η11=1, the measure ν := η1 defines a quasi-stationary law U. It follows that if (iii) is satisfied then there is also a quasi-stationary law U, which by Corollary 5.3 in [13] implies that the condition (i) in Theorem 1 holds. We will next justify the second part of Theorem 1. We assume that (i) holds βξ1 for some β>0. As we mentioned before, the set C = {β ∈ R : E(e 1{1<ζ}) < ∞}, is convex, it contains the element 0, and hence the whole interval [0,β]. It follows

128 V´ICTOR RIVERO

that (i) holds for any 0 <β <β.We denote by ηβ and ηβ the associated entrance laws for X. By (iii) we know they can be taken to be such that β β η1 1=1=η1 1. In order to get the claimed identities we make a short digression to recall further results obtained in [13]. There, it has been proved that whenever there is a θ>0 such that E(eθ(αξ1), 1 <ζ) ≤ 1, then there is a unique in law random variable Rθ such that, if it is taken independent of I, then Law IRθ = Zθ, where Zθ follows a Pareto distribution with parameter θ, viz. θ P(Z ∈ dy)= dy, y > 0. θ (1 + y)1+θ Besides, is worth noticing that elementary properties of the Beta and Gamma distributions imply that if 0 <θ <θand Bθ,θ−θ is an independent (θ ,θ− θ )– Beta random variable, viz.

Γ(θ) θ−1 (θ−θ)−1 P(B − ∈ dy)= y (1 − y) 1{ }dy, θ ,θ θ Γ(θ − θ)Γ(θ) 0

Law Zθ Zθ = . Bθ,θ−θ We deduce therefrom the identity in law

Law Rθ Rθ = . Bθ,θ−θ This being said, we apply these facts to θ = β/α and θ = β/α, with 0 <β <β. β β We should next relate the factors Rβ/α and Rβ/α with η and η , respectively. β β Let Jβ and Jβ be random variables with law η1 and η1 , respectively. Arguing β β as in page 482 in [13], raplacing ν there by η1 and η1 ,itisprovedthatifthese random variables are taken independent of I under P, then Law Law α Z α Z IJβ = β IJβ = β. Law Law α α Which implies Rβ = Jβ ,Rβ = Jβ , and

Law Jβ J = . β B1/α β β−β α , α With this information and the identity (8) it is easily veriﬁed that , / − β β β β α P B − ∈ − ηs f = s β , β β dz ηs Hz 1/α f, (0,1) α α for any f positive and measurable. This is what the identity (4) states. The constant Cβ,β in that equation appears when we remove the condition that the entrance law is such that the measure with time index 1 is a probability measure. Observe that the above argument also proves that for each γ>0thereisatmost

ENTRANCE LAWS FOR POSITIVE SELF-SIMILAR MARKOV PROCESSES 129 one γ-ssel, up to a multiplicative constant, which is given by the formula in (ii) in Theorem 1.

3. A pathwise extension of the identity (4) Our aim in this section is to provide a pathwise explanation of the curious identity (4). For that end we will carryout the following program. We will construct a self-similar process, X(β), associated to the entrance law ηβ, constructed in (ii) in Theorem 1. Then, for any pair 0 <β <βwe will relate, via a time change, the corresponding processes X(β ) and X(β). We will see that we can embed the paths of X(β ) into those of X(β). For this end, it will be necessary to assume throughout this section that (i) in Theorem 1 is satisfied with some index 0 <β<α.Under these assumptions, the papers [26]and[11] ensure the existence of a recurrent extension X(β) of X. That is, a process for which the state 0 is a recurrent and regular state, and such that X(β) killed at its first hitting time of 0 has the same law as X. We denote by N (β) the Itô’s excursion measure from 0 for X(β). The measure N (β) satisfies (i) N (β) is carried by the set of paths + {ω ∈ D | T0(ω) > 0andXt(ω)=0, ∀t ≥ T0};

(ii) for every bounded measurable f :[0, ∞) → R and each t, s > 0andΛ∈Gt (β) ∩{ } (β) E ∩{ } N (f(Xt+s), Λ t0. (β) AccordingtotheresultsinLemma2in[25], the entrance law (Ns ,s > 0) is a β-ssel for X. From Theorem 1, this is, up to a multiplicative constant, equal to that in (ii) in the op. cit. Theorem. In the case where Cramér’s condition is satisfied, (β) E βξ1 (e , 1 <ζ)=1, we have that limt→0+ Nt 1(a,∞) =0, for all a>0, in which case we say that the recurrent extension leaves 0 continuously. Whilst in the case (β) E βξ1 (e , 1 <ζ) < 1, we have limt→0+ Nt 1(a,∞) > 0, for all a>0, we say that the recurrent extension leaves 0 by a jump. In this case there is a jumping in measure η such that η(dx)=x−1−γ/αdx, x > 0, and dx η (dy)= P (X ∈ dy, t < T ),y>0,t>0. t 1+γ/α x t 0 (0,∞) x Moreover, in the paper [25] the process X(β) is constructed using Itô’s synthesis theorem. In what follows we will sketch the construction of another version of X(β), which will be such that its excursions are colored. We take a new process X taking values in [0, ∞) ×{−1, 1}, for which {0}× {−1, 1} is identified to a cemetery state, and it is such that when issued from (x, y) ∈ (0, ∞)×{−1, 1}, Xt =(Xt,y),t≥ 0, and X is issued from x. We understand X as a colored particle that moves, following the same dynamics as X, and that take the color red, for 1, and blue, for −1; and once the color is chosen it remains

130 VÍCTOR RIVERO fixed until the absortion at zero of the particle. The semigroup of X, say (Pt,t≥ 0), equals Pt((x, y),da⊗ dz)=Px(Xt ∈ da, t < T0) ⊗ δy(dz), (x, y), (a, b) ∈ [0, ∞) ×{−1, 1},t≥ 0. (β) D+ ×{− } (β) (β) ⊗ 1 Let N be the measure on 1, 1 defined by N = N ( 2 δ1(dx)+ 1 (β) 2 δ−1(dx)). Observe that N bears similar properties to those in (i)-(iii) above. + Realize a marked Poisson point process Δ = ((Δs,Us),s>0) on D ×{−1, 1} (β) with characteristic measure N . Thus each atom (Δs,Us)isformedofapath (Δs) and its mark Us. The marks are independent and follow a symmetric Bernoulli distribution. We will denote by T0(Δs,Us) the lifetime of the path (Δs,Us), i.e.

T0(Δs,Us) = inf{t>0:Δs(t)=0}. Set σt = T0(Δs,Us),t≥ 0. s≤t Since (β) −T (β) −T N (1 − e 0 )=N (1 − e 0 ) < ∞ it follows that for every t>0,σt < ∞ a.s. It follows that the process σ =(σt,t≥ 0) is an increasing càdlàg process with stationary and independent increments, i.e. a subordinator. Its law is characterized by its Laplace exponent φβ, defined by − − E(e λσ1 )=e φβ (λ),λ>0, and φ (λ) can be expressed thanks to the Lévy–Kintchine’s formula as β −λs φβ(λ)= (1 − e )νβ(ds), with νβ a measure such that s ∧ 1 νβ(ds) < ∞, called the Lévy measure of σ;see e.g. Bertoin [2] § 3 for background. An application of the exponential formula for Poisson point process gives − − − − (β) − λT0 − (β) − λT0 E(e λσ1 )=e N (1 e ) = e N (1 e ),λ>0, (β) −λT i.e. φβ(λ)=N (1 − e 0 ) and the tail of the Lévy measure is given by (β) νβ(s, ∞)=N (s0.

(β) −T0 Observe that if we assume φβ(1) = N (1 − e )=1thenφβ is uniquely deter- (β) mined. Since N has inﬁnite mass, σt is strictly increasing in t. Let Lt be the local time at 0, i.e. the inverse of σ

Lt =inf{r>0:σr >t} =inf{r>0:σr ≥ t}. (β) ≥ ≥ ≤ ≤ Deﬁne a process (Xt ,t 0) as follows. For t 0, let Lt = s, then σs− t σs, set − (β) (Δs(t σs−),Us)ifσs− <σs (10) Xt = (0, 0) if σs− = σs or s =0. That the process so constructed is a Markov process taking values in [0, ∞) × {−1, 0, 1} is a consequence of the main results in [5]and[30]. The arguments needed to check that the hypotheses in these papers are satisﬁed are an elementary

ENTRANCE LAWS FOR POSITIVE SELF-SIMILAR MARKOV PROCESSES 131 extension of those given in the paper [25] to verify that the corresponding conditions (β) are satisfied by X. We will denote by P its law. It is easily verified, from the construction, that the projection of X (β) in its first coordinate, is a version of the self-similar Markov process X(β).Forβ <β, we will next construct a version of X(β ) from X (β). The rest of this section will use facts from the fluctuation theory of Lévy processes, we refer to [1] for background on this topic. Fix q>0, and let A+,A−,q be the additive functionals of X (β) defined by t t + −,q α/β ≥ At = 1{X (β)∈(0,∞)×{1}}ds, At = q 1{X (β)∈(0,∞)×{−1}}ds, t 0, 0 s 0 s and we introduce the time change τ (q), which is the generalized inverse of the fluctuating additive functional A+ − A−,q, that is (q) { + − −,q } ≥ ∅ ∞ τ (t) = inf s>0:As As >t ,t0, inf = . Now, let Y (β) to be the projection of the process X (β) in the first coordinate and Y (+,q) be the process Y time changed by τ (q), (q) Y (q) if τ (t) < ∞, Y (+,q) = τ (t) t Δifτ (q)(t)=∞, where Δ is a cemetery or absorbing state. Notice that the time change has the effect of deleting all the blue paths in X (β), together with some red paths. So, the process, Y (+,q), is obtained by pasting together red paths, to which we have deleted a random length in its starting part, and the distribution of the deleted length depends on q. A more precise description is given in the following Lemma.

(β) Lemma 1. Under P the process Y (+,q), is a positive α-self-similar Markov process for which 0 is a regular and recurrent state and that leaves 0 by a jump according to the jumping-in measure cα,β,ρηρβ, with

−1−ρβ ηβρ(dx)=x dx, x > 0, where ρ is given by " " ## 1 α 1 − q πβ ρ = + arctan tan ∈]0, 1[, 2 πβ 1+q 2α

β(1−ρ) (β) βρ ∞ and 0

132 V´ICTOR RIVERO path of the excursion process in (0,r], conditioned to live for a period of time of length at least r. So the law of the meander at time 1 of X(β) is

(β) N (X1 ∈ dy|1

Similarly, that of Y +,q Law= X(βρ), is given by

(βρ) N (X1 ∈ dy|1 0), with ρ as deﬁned in Lemma 1,andL´evy measure

+ −,q ΠZ (dx)=π (dx)1{x>0} + π (−dx)1{x<0}.

(iii) The upward and downward ladder height processes, H and H,) associated β(1−ρ) to Z are stable subordinators of parameter βρ/α and α , respectively. −1 (β) Proof. Because the stable subordinator L has the L´evy measure N (T0 ∈ (β) ∈ − (β) −1−β/α dt), it follows that N (T0 dt)= dηt 1=ct dt, t > 0, for some constant 0 0) whose mark equals 1 and −1, respectively. The self-similarity

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(β) property for (Z+,Z−,q) follows from that of (X (β), P ). The jump measure π+ of Z+ is given by 1 π+(dt)= N (β)(T ∈ dt), 2 0 while that of Z−,q is q π−,q(dt)= N (β)(T ∈ dt). 2 0 This is a consequence of the following calculations based on the exponential formula for Poisson point processes: E(β) {− −,q} − − −λqα/β t (β) ∈ − exp λZ1 =exp (1 e )N (T0 dt, U = 1) (0,∞) −λqα/β t 1 (β) =exp − (1 − e ) N (T0 ∈ dt) (0,∞) 2 α/β c dt =exp − (1 − e−λq t) 1+ β (0,∞) 2 t α −λs q (β) =exp − (1 − e ) N (T0 ∈ ds) , (0,∞) 2 for all λ ≥ 0. The proof of the assertion in (ii) is straightforward. It is well known in the fluctuation theory of Lévy processes that the upward and downward ladder height subordinators associated to a stable Lévy process have the form claimed in (iii). See for instance [1] Chapter VIII.

Proof of Lemma 1. By construction, the closure of the set of times at which the process Y (+,q) visits 0 is the regenerative set given by the closure of the image of the supremum of the stable L´evy process Z. This coincides with the image of the upward ladder height subordinator H, which is a βρ/α-stable subordinator. This set is an unbounded perfect regenerative set with zero Lebesgue measure, see [2] Chapter 2. It follows that 0 is a regular and recurrent state for Y (+,q). The length of any excursion out of 0 for Y (+,q) is distributed as a jump of Z to reach a new supremum or, equivalently, as a jump of the upward ladder height process H associated to Z. Let N denotes the measure of the excursions from 0 of Z − Z, the process Z reﬂected at its current supremum, viz.

(Z − Z)t := sup {0 ∨ Zs}−Zt,t≥ 0; 0≤s≤t and let R denote the lifetime of the generic excursion from 0 of Z − Z.LetΠZ be the Lévy measure of Z and V) be the renewal measure of the downward ladder height subordinator H,) that is " ∞ # ) V (dy)=E 1 ds . {Hs∈dy} 0 It is known in the fluctuation theory for Lévy processes that under N the joint law of ZR− and ZR − ZR− is given by ) N(ZR− ∈ dx, −(ZR − ZR−) ∈ dy)=V (dx)ΠZ (dy)1{0

134 V´ICTOR RIVERO

Z R

Z R−

+1 -1 -1 +1 +1 +1 deleted path

Figure 1. A schematisation of the path of the processes X (β), and A+ − A−,q. The dots show the jumps of the process Z. The shaded area represents the portion of path that is deleted with the time change.

see for instance [18], Chapter VII. Moreover, the L´evy measure of H, say ΠH (dx), is such that ) ΠH ]x, ∞[= V (ds)ΠZ (du)1]x,∞[(u − s),x>0. {0≤s≤u} cf. [33]. In our framework, −(ZR − ZR−) denotes the length of the generic positive (β) excursion from 0 for (Y,P )andZR− is the length of the portion of the generic (β) positive excursion from 0 of (Y,P ) that is not observed while observing a generic (+,q) excursion from 0 of Y . See Figure 1. Furthermore, −(ZR − ZR−) − ZR− is the (+,q) β,+,q length of the generic excursion from 0 for Y , so N (T0 ∈ dt)=ΠH (dt). (β) Let N (·|T0 = ·) denote a version of the regular conditional law of the generic (β) β,+,q excursion under N given the lifetime T0. Similarly, the notation N (·|T0 = ·) will be used for the analogous conditional law under N β,+,q. These laws can be constructed using the method in [7], see also [25]. Finally, it follows from the verbal description above, that for any positive and measurable function f : R → R+ ∞ β,+,q β,+,q β,+,q N (f(Y0)) = N (T0 ∈ du)N (f(Y0)|T0 = u) 0 (β) = N(ZR− ∈ dt, −(ZR − ZR−) ∈ ds)N t∈]0,∞[ s∈]t,∞[ · (f(Y )|T = s, U =1) t 0 1 = V)(dt) Π (ds)N (β)(f(Y )|T = s, U =1) 2 Z t 0 t∈]0,∞[ s∈]t,∞[ ) (β) (β) = V (dt) N (T0 ∈ ds, U =1)N t∈]0,∞[ s∈]t,∞[ · (f(Y )|T = s, U =1) t 0 ) (β) = V (dt)N (f(Yt),t

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Given that the downward ladder height subordinator H) is a stable process with − index β(1 ρ) , it follows that α − − β,+,q β(1 ρ)) β(1 ρ) −1 (β) N (f(Y0)) = k dtt α N (f(Yt),t

= cα,ρϑηρf, where )k is a constant that depends on the normalization of the local time at zero for the reﬂected process Z − Z; which, without loss/ of generality can, and is supposed ) (β) βρ to be k =1. The ﬁniteness of N Y1 , 1

Remark 1. The previous proof is inspired in [28].

4. Two extensions In this section we state without proof two results that can be obtained with essentially the same proof as that of Theorems 1 and 2. 4.1. A result in the fluctuation theory of self-similar Markov processes. This subsection is motivated by the work [8]. Let (Z, h)beabivariate Lévy process such that its coordinates are subordinators. Denote by ΠZ , Πh the Lévy measures of Z and h, respectively. We will use the following notation for tail Lévy measures x ΠZ (x)=ΠZ (x, ∞), Πh(x)=Πh(x, ∞),Ah(x)=max{1, Πh(1)} + Πh(z)dz, 1 x>0. For α>0, let τ be the time change defined as h s αhs τh(t) = inf{s>0: e ds > t},t>0. 0

hs Deﬁne a stochastic process (Vt,t≥ 0) pathwise as the Stieltjes integral of e with respect to Z, t αhs− Vt = e dZs,t≥ 0. 0

136 V´ICTOR RIVERO

For a ≥ 0,x>0wedenotebyQa,x the law of the processs (R, H) deﬁned as follows −α τh(tx ) α αhs− hτ (tx−α) Rt = a + x e dZs,Ht = xe h ,t≥ 0. 0

Theorem 3. Under (Qa,x,a≥ 0,x>0) (R, H) is a Feller process in [0, ∞) × (0, ∞). Assume that ∞ log(y) (12) ΠZ (dy) < ∞. e Ah(log(y)) ∞ −αhs The random variable I := 0 e dZs− is finite a.s. The family of measures (μt,t≥ 0) defined by " # tI t 1/α 1 μtf = E f , , Ih Ih Ih ∞ −αhs form an entrance law for Q·,·, where Ih := 0 e ds. The condition (12) has been introduced by Linder and Maller [21], and it is ∞ −αhs− a necessary and sufficient for the Lévy integral, 0 e dZs, to be finite. Their results hold not only for subordinators but for any Lévy process. It is important to + mention that if E(h1) < ∞ then the condition (12) is equivalent to E(log (Z1)) < ∞ which is a well known equivalent condition for the convergence a.s. of the integral ∞ −s 0 e dZs, see e.g. [31]. It is important to mention that it is possible to establish a result similar to Theorem 3 for more general couples of Lévy processes but, as we do not have any application in mind, we wont pursue this line of research. In the work [8] the process h is the upward ladder height associated to a Lévy process ξ and Z is another subordinator constructed as functionals of the excursions of ξ reflected in its past supremum. This processes are key elements to develop a fluctuation theory for pssMp. We refer to [8] for further details. That the process (R, H)isaFellerprocessin[0, ∞) × (0, ∞), is obtained using arguments similar to those in [8]. The rest of the proof follows along the same lines of Theorem 1; mainly the roll played by the Lévy process ξ is played by the bivariate 2 Lévy process (Z, h). It is known that Lebesgue measure Λ2 in R is an invariant measure for any bivariate Lévy process, so for (Z, h). We take (Y =(Y1,Y2), Q) the Kuznetzov process associated to (Z, h)andΛ2. This process has the Markov property under Q and is invariant under translations. The birth and death time of Y under Q are infinite because the Lebesgue measure is invariant for (Z, h)and (Z,) )h)=−(Z, h). The rest of the proof is essentially the same as that of Theorem 1. 4.2. Multi-self-similar Markov processes. The following definition stems from the work of Jacobsen and Yor [16]. Definition 1 (Jacobsen and Yor [16]). A n-dimensional Markov process X Rn ∞ n ∈ with state space + =[0, ) is 1/α-multi-self-similar, with α =(α1,...,αn) n R , if for all scaling factors c1,...,cn > 0, and all initial states x =(x1,...,xn) ∈ Rn + it holds that ""(13) # # ,, / / i ≥ P Law i ≥ P ciXt/c ,t 0 , (x1,...,xn) = Xt ∈ ,t 0 , (c1x1,...,cnxn) , i∈(1,...,n) i (1,...,n) * n αi { i ∈ where c = i=1 ci . We denote by T0 =inft>0:Xt =0, for some i {1,...,n}}.

ENTRANCE LAWS FOR POSITIVE SELF-SIMILAR MARKOV PROCESSES 137

Note that for n>1, the symbol 1/α is senseless, but we made the choice of using it to preserve the customary notation for the 1-dimensional case. Of course, in the case n =1, 1/0 neither makes sense but observe that in the definition (13) this does not cause any inconvenience. Examples of multi-self-similar diffusions where introduced by Jacobsen [15]and Jacobsen and Yor [16]. Examples of processes with jumps can be easily obtained from these processes by subordination via independent stable subordinators. Extending the arguments of Lamperti [20], Jacobsen and Yor [16] established that there exists a one to one correspondence between multi–self–similar Markov Rn Rn ∪{ } processes on + and Lévy processes taking values in Δ , that we next sketch. For that end we start by setting some notation.* ∈ Rn n αi ∈ For α fixed, we will denote pα(u)= i=1 ui , for all u =(u1,...,un) (0, ∞)n. For u, v ∈ [0, ∞)n we denote by u ◦ v the Hadamard product of u and v, n u ◦ v =(u1v1, ··· ,unvn). For a vector z =(z1,...,zn) ∈ R ∪{Δ} we will denote E { } { } ∈ Rn E ∈ { ∈ ∞ n by* (z)=(expz1 ,...,exp zn ) , if z and (Δ) 0 := u [0, ) : n } D D ∞ → i=1 ui =0. Assume now that ( , )isthespaceofcàdlàg paths ω :[0, [ Rn ∪{Δ}, endowed with the σ–algebra generated by the coordinate maps and the completed natural filtration (Dt,t≥ 0). Let P be a probability measure on D such that under P the process ξ is a Lévy process that takes values in Rn ∪{Δ}. Where the state Δ is understood as a cemetery state, and so the first hitting time of Δ or life time for ξ, say ζ, follows an exponential distribution with some parameter q ≥ 0, and the value q =0, is permitted to include the case where ζ = ∞, P-a.s. For α ∈ Rn, set for t ≥ 0 s τ(t) = inf{s>0, e<α,ξr >dr > t}, 0 {∅} ∞ ∈ ∞ n x ≥ with the usual convention that inf = . For x [0, ) , let (Xt ,t 0) be the process defined by Lamperti, Jacobsen and Yor transformation: x ◦ E ξ , if t

}ds (14) X(x) := τ(t/pα(x)) α 0 s t ≥ ζ { } ∈ 0ift pα(x) 0 exp < α,ξs > ds or x 0.

n (x) for t ≥ 0. For x ∈ [0, ∞) , we denote by Px the law of X , that is the image measure of P under Lamperti’s transformation applied to the process ξ.Itisa standard fact that this process is adapted with respect to the filtration Ft = Dτ(t), t ≥ 0, and inherits the strong Markov property from ξ, with respect to the filtration (Ft,t ≥ 0), see for instance [29]. A straight forward verification shows that the Markov family (X, Px)x∈[0,∞)n bears the 1/ α-multi-self-similar property. Jacobsen and Yor proved that any multi-self-similar Markov process that never hits the set 0, can be constructed this way. A perusal of their proof and that of Lamperti for the case of dimension 1, shows that the result can be easily extended to any multi- self-similar Markov process killed at its first hitting time of 0. We do not include the details. We have the following result that extends the Theorem 2 for the class of multi-self-similar Markov processes. , / n Theorem 4. Let α =(α ,...,α ) ∈ R , X = (X ) ≥ , (P ) ∈R+ be a 1/ α- 1 n t t 0 x x n multi-self-similar Markov process and ξ the Rn-valued Lévy process associated to it via the Lamperti-Jacobsen-Yor transformation. Assume that ξ has an infinite lifetime and limt→∞ <α,ξt >= ∞. There exists a unique entrance law (μt,t ≥ 0)

138 V´ICTOR RIVERO for X whose λ-potential is given by ∞ −λt dte μtf 0

= m(dx1 ···dxn)f(x1, ··· ,xn)E (0,∞)n " ∞ #

· exp{−λpα ((x1,...,xn)) exp{− < α,ξs >}ds} , 0 where f :(0, ∞)n → [0, ∞) is a measurable function and &n αi−1 n m(dx1 ···dxn)=dx1 ···dxn (xi) , on (0, ∞) . i=1 * n αi This entrance law has the scaling property: for c1,...,cn > 0,c= i=1 ci

(15) μt/cH(c1,...,cn)f = μtf, t > 0, with ∈ ∞ n H(c1,...,cn)f(x1,...,xn)=f(c1x1,...,cnxn), (x1,...,xn) (0, ) The proof of this result follows essentially the same argument as that of Theo- rem 1. For, it is necessary ﬁrst to recall that Lebesgue’s measure in Rn, say Λ, is an invariant measure of ξ, and second that ξ and ξ) = −ξ, are in weak duality with respect to it. Then we construct the Kusnetzov process associated to Λ. This measure is invariant under translations. The rest of the proof is similar. The scaling property can be veriﬁed as in the proof of Theorem 1 using the invariance under translations.

References [1] Jean Bertoin, Lévy processes, Cambridge Tracts in Mathematics, vol. 121, Cambridge Uni- versity Press, Cambridge, 1996. MR1406564 (98e:60117) [2] Jean Bertoin, Subordinators: examples and applications, Lectures on probability theory and statistics (Saint-Flour, 1997), Lecture Notes in Math., vol. 1717, Springer, Berlin, 1999, pp. 1– 91. MR1746300 (2002a:60001) [3] Jean Bertoin and Maria-Emilia Caballero, Entrance from 0+ for increasing semi-stable Markov processes, Bernoulli 8 (2002), no. 2, 195–205. MR1895890 (2003c:60071) [4] Jean Bertoin and Marc Yor, Theentrancelawsofself-similarMarkovprocessesandex- ponential functionals of Lévy processes, Potential Anal. 17 (2002), no. 4, 389–400, DOI 10.1023/A:1016377720516. MR1918243 (2003i:60082) [5] R. M. Blumenthal, On construction of Markov processes,Z.Wahrsch.Verw.Gebiete63 (1983), no. 4, 433–444, DOI 10.1007/BF00533718. MR705615 (84j:60081) [6] M. E. Caballero and L. Chaumont, Weak convergence of positive self-similar Markov processes and overshoots of Lévy processes, Ann. Probab. 34 (2006), no. 3, 1012–1034, DOI 10.1214/009117905000000611. MR2243877 (2008c:60035) [7] L. Chaumont, Excursion normalisée, méandre et pont pour les processus de Lévy stables (French, with French summary), Bull. Sci. Math. 121 (1997), no. 5, 377–403. MR1465814 (99a:60077) [8] Lo¨ıc Chaumont, Andreas Kyprianou, Juan Carlos Pardo, and V´ıctor Rivero, Fluctuation theory and exit systems for positive self-similar Markov processes, Ann. Probab. 40 (2012), no. 1, 245–279, DOI 10.1214/10-AOP612. MR2917773 [9] C. Dellacherie, B. Maisonneuve, and P. A. Meyer, Probabilités et potentiel: Processus de Markov (fin). Compléments du calcul stochastique, vol. V, Hermann, Paris, 1992. [10] S. Dereich, L. Doering, and A. E. Kyprianou, Real Self-Similar Processes Started from the Origin, ArXiv e-prints (2015), XX.

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[11] P. J. Fitzsimmons, On the existence of recurrent extensions of self-similar Markov processes, Electron. Comm. Probab. 11 (2006), 230–241, DOI 10.1214/ECP.v11-1222. MR2266714 (2008h:60143) [12] R. K. Getoor, Excessive measures, Probability and its Applications, Birkhäuser Boston, Inc., Boston, MA, 1990. MR1093669 (92i:60135) [13] Bénédicte Haas and V´ıctor Rivero, Quasi-stationary distributions and Yaglom limits of self- similar Markov processes, Stochastic Process. Appl. 122 (2012), no. 12, 4054–4095, DOI 10.1016/j.spa.2012.08.006. MR2971725 [14] Bénédicte Haas and V´ıctor Rivero, Factorizations of generalized Pareto random variables using exponential functionals of Lévy processes and applications, Work in progress (2014), xx. [15] Martin Jacobsen, Discretely observed diffusions: classes of estimating functions and small Δ-optimality, Scand. J. Statist. 28 (2001), no. 1, 123–149, DOI 10.1111/1467-9469.00228. MR1844353 (2002j:62108) Rn [16] Martin Jacobsen and Marc Yor, Multi-self-similar Markov processes on + and their Lamperti representations, Probab. Theory Related Fields 126 (2003), no. 1, 1–28, DOI 10.1007/s00440-003-0263-5. MR1981630 (2004g:60062) [17] H. Kaspi, Random time changes for processes with random birth and death, Ann. Probab. 16 (1988), no. 2, 586–599. MR929064 (89f:60080) [18] Andreas E. Kyprianou, Fluctuations of Lévy processes with applications, 2nd ed., Universi- text, Springer, Heidelberg, 2014. Introductory lectures. MR3155252 [19] John Lamperti, Semi-stable stochastic processes, Trans. Amer. Math. Soc. 104 (1962), 62–78. MR0138128 (25 #1575) [20] John Lamperti, Semi-stable Markov processes. I, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 22 (1972), 205–225. MR0307358 (46 #6478) [21] Alexander Lindner and Ross Maller, Lévy integrals and the stationarity of generalised Ornstein-Uhlenbeck processes, Stochastic Process. Appl. 115 (2005), no. 10, 1701–1722, DOI 10.1016/j.spa.2005.05.004. MR2165340 (2006i:60070) [22] Joanna B. Mitro, Dual Markov processes: construction of a useful auxiliary process,Z. Wahrsch. Verw. Gebiete 47 (1979), no. 2, 139–156, DOI 10.1007/BF00535279. MR523166 (80g:60075) [23] Henry Pant´ı, Juan Carlos Pardo, and V´ıctor Rivero, Recurrent extensions of real-valued self- similar Markov processes, Work in progress (2014), xx. [24] Juan Carlos Pardo and V´ıctor Rivero, Self-similar Markov processes,Bol.Soc.Mat.Mexicana (3) 19 (2013), no. 2, 201–235. MR3183994 [25] V´ıctor Rivero, Recurrent extensions of self-similar Markov processes and Cramér’s condition, Bernoulli 11 (2005), no. 3, 471–509, DOI 10.3150/bj/1120591185. MR2146891 (2006d:60118) [26] V´ıctor Rivero, Recurrent extensions of self-similar Markov processes and Cramér’s condition. II, Bernoulli 13 (2007), no. 4, 1053–1070, DOI 10.3150/07-BEJ6082. MR2364226 (2008k:60173) [27] V´ıctor Rivero, Tail asymptotics for exponential functionals of Lévy processes: the convolution equivalent case (English, with English and French summaries), Ann. Inst. Henri Poincaré Probab. Stat. 48 (2012), no. 4, 1081–1102, DOI 10.1214/12-AIHP477. MR3052404 [28] L. C. G. Rogers and David Williams, Time-substitution based on fluctuating additive functionals (Wiener-Hopf factorization for infinitesimal generators), Seminar on Probability, XIV (Paris, 1978/1979), Lecture Notes in Math., vol. 784, Springer, Berlin, 1980, pp. 332–342. MR580139 (82b:60089b) [29] L. C. G. Rogers and David Williams, Diffusions, Markov processes, and martingales. Vol. 1, 2nd ed., Wiley Series in Probability and Mathematical Statistics: Probability and Math- ematical Statistics, John Wiley & Sons, Ltd., Chichester, 1994. Foundations. MR1331599 (96h:60116) [30] Thomas S. Salisbury, Construction of right processes from excursions, Probab. Theory Re- lated Fields 73 (1986), no. 3, 351–367, DOI 10.1007/BF00776238. MR859838 (88g:60177) [31] Ken-iti Sato, Lévy processes and infinitely divisible distributions, Cambridge Studies in Ad- vanced Mathematics, vol. 68, Cambridge University Press, Cambridge, 1999. Translated from the 1990 Japanese original; Revised by the author. MR1739520 (2003b:60064) [32] Michael Sharpe, General theory of Markov processes, Pure and Applied Mathematics, vol. 133, Academic Press, Inc., Boston, MA, 1988. MR958914 (89m:60169)

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Centro de Investigacion´ en Matematicas´ (CIMAT A.C.) E-mail address: [email protected]

Contemporary Mathematics Volume 656, 2016 http://dx.doi.org/10.1090/conm/656/13106

Combinatorics and geometry

Fernando Rodriguez-Villegas

Abstract. The goal of this talk was to illustrate how the two concepts of the title, Combinatorics and Geometry, perhaps seemingly unrelated at ﬁrst sight, are really tied together quite closely.

The goal of this talk was to illustrate how the two concepts of the title, perhaps seemingly unrelated at ﬁrst sight, are really tied together quite closely. Combina- torics, the art of counting, naturally deals with discrete objects. Geometry on the other hand mostly evokes the opposite, the continuous, dealing with shapes in space and the like. But there is a tight connection between the two. We may even go further and speculate whether our universe is continuous or discrete. Arguably, the natural answer is the former but it increasingly it looks like it might just be the latter.

§1. What is a projective plane? We can abstract axiomatically the basic properties of what a projective should be: a set A whose elements we call points and a collection of subsets of A which we call lines satisfying a few simple axioms: • Two distinct points lie in a unique line. • Two distinct lines meet in a unique point. • There exist four points not all in a line. The first two axioms are the key features of a projective plane, the third avoids dealing with some trivial cases. The main point we would like to emphasize is that no finiteness is required: A could be a finite set. For example, Fano in 1892 produced a projective plane consisting of seven points and seven lines (see Fig.1). In general, in any finite projective plane all lines have the same number of points, say q + 1 points for some q. The total number of points of the plane is then q2 + q + 1, with q = 2 in Fano’s case (the smallest possible). The familiar theorem of Desargues of usual projective geometry (see Fig.2, the theorem says that the lines A, A, B,B and C, C necessarily meet at a point) does not follow from the above axioms. In fact, it holds if and only if we can give coordinates to our plane. If Desargues theorem holds we can use it to define all the usual operations with coordinates: sums, multiplication by scalars, etc. The scalars obtained inherit the algebraic structure of a field. For a finite plane we get n a finite field Fq, necessarily of size q = p elements for some prime number p.Up

2010 Mathematics Subject Classiﬁcation. Primary 11G25, 05A30.

c 2016 American Mathematical Society 141

142 FERNANDO RODRIGUEZ-VILLEGAS

Figure 1. Fano plane

Figure 2. Desargues theorem

(1:0:0)

(1:0:1) (1:1:0)

(1:1:1)

(0:0:1) (0:1:0) (0:1:1)

Figure 3. Fano plane coordinates to isomorphism this ﬁeld is uniquely determined by its size q. You can see in Fig. 3 the coordinates system in the Fano plane where the scalars consist of the ﬁeld of two elements F2 = {0, 1}.

§2. In algebraic geometry we study the zero locus X(C) of polynomials, say, F1,...,Fm with complex coefficients in variables x1,...,xn. If the coefficients of Fi are actually integers we may also consider their solutions X(Fq) with coordinates in the finite field Fq. What, if any, is the relation between the complex points X(C) and the finite field points X(Fq)?

COMBINATORICS AND GEOMETRY 143

Figure 4. X(C) genus 3 curve

Consider for example, X : y2 = f(x) with f ∈ Z[x] square-free of degree 8. This is an algebraic curve of genus g =3. Its complex points X(C) look like Figure 4. Topologically X(C) is a three-holed doughnut. What can we say about X(Fq)? Pictures of this ﬁnite set are not particularly useful. Passing from C to Fq we go from the continuous to the discrete and loose our ordinary spatial intuition. But we gain something else: we can count. Thanks to the work of Weil we know that for a smooth projective curve X of genus g we have for all n 2g 1 F n − n | | 2 #X( pn )=p +1 αi , αi = p i=1 Hence with q = pn √ (0.1) |#X(Fq) − q − 1|≤2g q.

In particular, if g =0then#X(Fq)=q + 1. This is as it should be! Indeed, X is isomorphic to a projective line, which as we discussed has q + 1 points if the field of scalars is Fq. We can interpret the inequality (0.1) as saying that a general algebraic curve of F genus√g has roughly q +1 pointsover q, the points in a line, with an error√ bounded√ by 2g q. This error does indeed range over the whole interval [−2g q, 2g q]as X and q vary. Hence, on one hand g determines the topological shape of X(C)and on the other it controls the rough behaviour of #X(Fq). In general, what precisely can we recover of X(C)fromthe#X(Fq) data? In a loose analogy the situation is similar to that of tomography. We may think of passing to a given finite field as analogous to taking the sectional image of an object in space along a given plane. In tomography one reconstructs the shape of object from the sectional images along all planes. One may hope that data on X(Fq)would allow the recovery of the shape of X(C). To some extent this is the case (see [4] and [2] for two well know and important examples) thanks to the Weil conjecures proved by Deligne. A particular, simple situation is the following. Suppose X(C) is smooth, com- j pact and #X(Fq)=C(q) for a certain polynomial C.Letbj (X):=dimH (X, C), the Betti numbers of X. (These are topological invariants of the space X(C); for

144 FERNANDO RODRIGUEZ-VILLEGAS example, for an a algebraic curve of genus g we have b0 = b2 =1,b1 =2g and all others are zero.) Then b2i+1(X)=0and dimX i (0.2) C(q)= b2i(X) q i=0 For example, if X = P1, the projective line, then C(q)=q +1 2 and indeed we have b0 = b2 =1,b1 =0sinceg = 0. Similarly, if X = P ,the projective plane, then as we mentioned in §1 C(q)=q2 + q +1 and indeed b0 = b2 = b4 =1,b1 = b3 =0. We must point out a technical but crucial point for what follows. The equality (0.2) holds in a bit more generality. It is enough to know that the natural mixed Hodge structure in the cohomology of X is pure and #X(Fq)=C(q) or all finite fields with q = pn for all but finitely many primes p. Note that such counting polynomials C(q) must have non-negative integer coefficients (these coefficients being dimensions of vector spaces). The relation (0.2) is actually a two-way street. We may use it to compute Betti numbers by counting or to prove that certain polynomials have non-negative coefficients (because they happen to equal C(q) for an appropriate X). We discuss an example along the lines of the latter situation in §3below. Consider for example the Grassmanian X = G(k, n) of all dimension k subspaces in a fixed n dimensional space. Its number of points is given by the q- binomial coefficient $ % n [n]! #G(k, n)(F )= = , [n]! := (1 − q) ···(1 − qn), q k [k]![n − k]! which is a polynomial in q. E.g., $ % 5 = q6 + q5 +2q4 +2q3 +2q2 + q +1 2 and therefore for X = G(2, 5) we have b0 = b2 = b10 = b12 =1,b4 = b6 = b8 =2 and all other Betti numbers are zero.

§3. A quiver Q is a directed graph. A representation of Q is an assignment: vertex → vector space arrow → linear map We are interested in representations up to isomorphism. For example, if Q is the quiver Sg (see Fig. 5) consisting of one vertex with g loops attached then a representation is a tuple (A1,...,Ag)ofn × n matrices for some n. Two representations (A1,...,Ag)and(A1,...,Ag) are isomorphic if the tuples of matrices are simultaneously conjugate. I.e. −1 (A1,...,Ag)=U(A1,...,Ag)U for some invertible matrix U.Forg = 1 this is Jordan’s problem: to classify matrices up to conjugation; it has a beautiful solution that we learn in a linear algebra course.

COMBINATORICS AND GEOMETRY 145

Figure 5. Sg quiver

Can we classify in some form representations up to isomorphism in general? For example, can we classify g>1 tuples of matrices up to simultaneous conjugation? Mostly we cannot; these are typically difficult linear algebra problems. Kac (in the early 80’s) thought of passing to finite fields and counting representations up to isomorphism. Fix Q and a dimension vector α (recording the dimension of the vector spaces attached to the vertices of Q). Consider absolutely indecomposable representations of dimension α. These are representations that do not decompose in a non-trivial way as a direct sum even after extending scalars to a larger field. In Jordan’s case an absolutely indecomposable representation of dimension n is given by an n×n matrix with exactly one Jordan block. To illustrate the issue of extending scalars, note that for example the matrix " # 01 −10 is not absolutely indecomposable. It represents an indecomposable representation over R but not over C since its Jordan decomposition is diagonal with eigenvalues i and −i. Kac showed that up to isomorphism the number of absolutely indecomposable representations of a quiver Q of dimension α equals Aα(q) a polynomial in q with integer coefficients. It is not a particularly easy polynomial to compute though there is a rather daunting formula for a certain generating function of these [6]. For example, for the Sg quiver we have α\g 1 2 3 4 1 q q2 q3 q4 2 q q5 + q3 q9 + q7 + q5 q13 + q11 + q9 + q7 3 q q10 + q8 + q7 + ··· q19 + q17 + q16 + ··· q28 + q26 + q25 + ··· 4 q q17 + q15 + q14 + ··· q33 + q31 + q30 + ··· q49 + q47 + q46 + ··· 5 q q26 + q24 + q23 + ··· q51 + q49 + q48 + ··· q76 + q74 + q73 + ··· 6 q q37 + q35 + q34 + ··· q73 + q71 + q70 + ··· q109 + q107 + q106 + ···

Note that for g =1wehaveAα(q)=q for all α since, as mentioned, an absolutely indecomposable representation of dimension α consists of one Jordan block of size α. This Jordan block is uniquley determined up to isomorphism by its eigenvalue for which there are q = |Fq| possibilities. Kac conjectured that the coeﬃcients of Aα(q) are in fact non-negative. Crawley- Boevey and van der Bergh proved the conjecture when α indivisible (not a proper

146 FERNANDO RODRIGUEZ-VILLEGAS

Figure 6. Extended Sg quiver

multiple of another integral vector). For the quivers Sg, for example, it only applies to the case of dimension 1. With Hausel and Letellier we extended the proof to the general case, see §4. The argument of Crawley-Boevey and van der Bergh shows that, in fact, − 2i Q C i dα/2 (0.3) Aα(q)= dim Hc ( α; ) q , i where Qα is an associated smooth Nakajima quiver variety of dimension dα.The hypothesis on α being indivisible is crucial for the existence of Qα. This variety is an appropriate replacement for a naive “space of isomorphism classes” of representations of the type we want to count and the identity (0.3) is far from obvious. The proof of (0.3) is along the lines of our discussion in §2. Namely,

dα/2 #Qα(Fq)=q Aα(q).

However, this is not quite enough to deduce (0.3) because though Qα is smooth it is not compact. A further argument is needed to show that (0.2) still holds for Qα because the natural mixed Hodge structure on its cohomology is pure.

§4. Given a quiver Q and arbitrary dimension vector α with Hausel and Letellier (see [5] and the references given therein) we consider an extended quiver Q˜ adding legs to every vertex. On Q˜ we take the dimension vectorα ˜ where if αi is the dimension at the i-th vertex of Q then we put dimensions αi − 1,αi − 2,... along the vertices of the attached leg. For example, if Q = Sg and α = n then Q˜ is as in Fig.6 with a leg of length n andα ˜ = n, n − 1,...,2, 1. The vectorα ˜ is indivisible and we may hence consider the associated Nakajima quiver variety Q˜α˜ of dimension dα˜. Each vertex of the quiver gives rise to a reflection that acts on the cohomology of Q˜α˜. The reflections along the i-th leg generate a × ··· copy of the symmetric group Sαi and we therefore obtain an action of Sα1 Sα2 . × ··· Let = 1 2 where i is the sign character of Sαi . Our main result is that − 2i Q C i dα˜ /2 Aα(q)= dim Hc ( α˜; ) q , i where the subscript indicates we take the isotypical component of the corresponding space. This again is proved by counting points over finite fields using the whole

COMBINATORICS AND GEOMETRY 147 machinery for counting points on character and quiver varieties developed in our previous papers.

References [1] William Crawley-Boevey and Michel Van den Bergh, Absolutely indecomposable representations and Kac-Moody Lie algebras, Invent. Math. 155 (2004), no. 3, 537–559, DOI 10.1007/s00222-003-0329-0. With an appendix by Hiraku Nakajima. MR2038196 (2004m:17032) [2] Lothar Göttsche, The Betti numbers of the Hilbert scheme of points on a smooth projective surface, Math. Ann. 286 (1990), no. 1-3, 193–207, DOI 10.1007/BF01453572. MR1032930 (91h:14007) [3] Victor G. Kac, Root systems, representations of quivers and invariant theory, Invariant theory (Montecatini, 1982), Lecture Notes in Math., vol. 996, Springer, Berlin, 1983, pp. 74–108, DOI 10.1007/BFb0063236. MR718127 (85j:14088) [4] G. Harder and M. S. Narasimhan, On the cohomology groups of moduli spaces of vector bundles on curves, Math. Ann. 212 (1974/75), 215–248. MR0364254 (51 #509) [5] Tamás Hausel, Emmanuel Letellier, and Fernando Rodriguez-Villegas, Positivity for Kac polynomials and DT-invariants of quivers, Ann. of Math. (2) 177 (2013), no. 3, 1147–1168, DOI 10.4007/annals.2013.177.3.8. MR3034296 [6] Jiuzhao Hua, Counting representations of quivers over finite fields,J.Algebra226 (2000), no. 2, 1011–1033, DOI 10.1006/jabr.1999.8220. MR1752774 (2001j:16017)

Univ. of Texas at Austin – and – ICTP Trieste E-mail address: [email protected] E-mail address: [email protected]

Contemporary Mathematics Volume 656, 2016 http://dx.doi.org/10.1090/conm/656/13105

A (short) survey on dominated splittings

Mart´ın Sambarino Dedicated to Ricardo Ma˜n´e (1948-1995) and Jorge Lewowicz (1937-2014)

Abstract. In the last two decades, dominated splitting has been playing a central role in the theory of global dynamics, particularly understanding non- hyperbolic dynamics and extending many results known for hyperbolic theory. We present here this notion, its basic properties, a brief account on its relation- ship with global dynamics, and give some important results on its dynamical consequences and shed some light on how it can fail to be hyperbolic and related problems. The complete understanding of these results and problems will represent a deﬁnite step to have a comprehensive picture of global dynamics.

1. Introduction Since the invention of Calculus a basic tool in order to get information on a smooth object is to look to its “linear approximation”. Let me start with two very simple examples to give a rough idea of the purpose of the present paper. First, as we learn during the first courses, when we have a smooth function of the real line or the interval, if we know that at some point x, the derivative f (x) has a definite sign, then we know that the function is strictly monotonous on a neighborhood of the point x. Notice that any map g smoothly close to f will have g(x) of the same sign. On the other hand if f (x) = 0 then we cannot ensure what the behavior of f around x will be. However, if we know that on a C1-neighborhood of f, all maps have the same behavior around a uniform neighborhood of x, then we conclude that the derivative of f has a definite sign at x. The second example I would like to mention comes from the study of autonomous differential equations. Let x = f(x) be a differential equation where f :Ω→ Rn is a smooth map on an open region Ω. Suppose that we have an isolated equilibrium point x0 and let A = Dfx0 . We know that if all the eigenvalues of A have negative real part then the equilibrium point x0 is (asymptotically) stable. Notice that if g is smoothly close to f, the corresponding differential equation will have an asymptotically stable equilibrium point near x0. On the other hand,

2010 Mathematics Subject Classification. Primary 37D30, 37-06, 37-02. Ricardo Mañé was an Uruguayan mathematician that started his studies in dynamical systems with Jorge Lewowicz and developed his career at IMPA. His numerous and remarkable contributions are always inspiring. I first him at Lewowicz’s place in Montevideo. Jorge Lewowicz is the father of the dynamical systems school in Uruguay and passed away in June 2014. We deeply miss his encouragement, profoundly insights, conversations, friendship and sense of humor.

c 2016 American Mathematical Society 149

150 MARTÍN SAMBARINO if all eigenvalues of A have non positive real part (but there is one with zero real part) then we cannot ensure what the behavior around x0 will be. However, if we know that for any g on a C1 neighborhood of f, the corresponding differential equation has a unique equilibrium point in a uniform neighborhood of x0 and it is asymptotically stable, then all the eigenvalues of A have negative real part. I mentioned the above examples to support the following idea (or principle) in smooth dynamics, which is somehow the leitmotif of this survey: a“stable” structure of the linear approximation should allow one to describe the dynamics (or at least should impose restrictions on it) and a “stable” dynamical phenomena implies some “stable” structure on the linear approximation. This principle has been beautifully accomplished in the so called hyperbolic theory started by D.V. Anosov and S. Smale in the sixties and an endless list of contributors (Bowen, Franks, Katok, Mañé, Newhouse, Palis, Pugh, Sinai, Shub just to mention a few). Let us explain it in a very informal way. A smooth dynamical system f : M → M is hyperbolic on a compact set Λ and invariant (f(Λ) = Λ) if the tangent map Df has a hyperbolic structure over the set, meaning that s u the tangent bundle splits into two subbundles TΛM = E ⊕ E invariant under Df and the action of Df on this subbundles has a uniform behavior (uniformly contracting on Es and uniformly expanding on Eu). A precise definition is given in the next section. This hyperbolic structure of Df has strong implications on the dynamics of f. Locally, the dynamics splits into two “directions” (a family of two transversal invariant submanifolds called stable and unstable ones) where in one direction (stable) points get exponentially close by forward iteration and on the other one (unstable) points get exponentially close by backward iteration. These facts allow to successfully describe the dynamics (see for instance [S1], [Sh1], [R1], [KH]). On the other hand this hyperbolic structure cannot be destroyed by small smooth perturbations and “stability” can be obtained from here. However, hyperbolicity is less universal than it was originally thought and weaker forms of hyperbolicity appeared in the literature: Partial Hyperbolicity and Dominated Splitting. These (non hyperbolic) structures on the tangent map Df of a dynamical system f : M → M can not be destroyed by small perturbations. The weaker one having this property is dominated splitting. Roughly speaking, we say that an invariant set has dominated splitting if the tangent bundle over the set splits into two invariant subbundles which are invariant under Df and although we do not know (a priori) that the action of Df on these subbundles has a uniform behavior (contracting or expanding), we know that the action on one dominates the action on the other one (see Definition 2.2). This “domination” prevents the structure to be destroyed by small perturbations. This dominated splitting structure may allow us to think that the dynamics of f splits into two directions. Nevertheless, since the action of Df on the invariant subbundles might not have a uniform behavior we can not get information on how the dynamics on these “direction” would be: although a family of transverse locally invariant submanifolds do exist, we do not have information aprioriwhether they have a dynamical meaning. And so, trying to get dynamical information from the dominated splitting seems hopeless. Can we describe the dynamics of a set having dominated splitting? After a first glance, the question is very naive since any dynamical system f : M → M can be embedded into another one having dominated splitting (multiplying by other systems having strong contraction/expansion), though in this case the dynamics

DOMINATED SPLITTING 151 lives in lower dimensional submanifold. We have to take a deeper look to see which are the right questions to ask. Besides, can we characterize when a set having dominated splitting fails to be hyperbolic? We will see through the following pages there are many results in this direction though there is still a long way to a comprehensive understanding of it. Dominated splitting plays a central role in the understanding of global dynamics from a (C1) generic viewpoint. We will review some aspects of this theory in the next section, though we won’t be very exhaustive (I recommend to the interested reader the survey by C. Bonatti ([B]), the comprehensive work by Crovisier in [C1] and [C5]). Other related surveys that the reader could consult are [PS4], [P1], [P2]andthebook[BDV]). Many important topics concerning dominated splitting are not included in this survey for time and space reasons and I apologize for this. Let me mention just a few of them: • The study of C1 generic diffeomorphisms preserving a volume form that startedwiththeseminalworkofR.Mañé([M6]). As a reference to the subject see the paper by Avila and Bochi [AB]. Recently, Avila-Crovisier- Wilkinson proved a landmark result in this area [ACW]. It is also present in the study of stable ergodicity for diffeomorphisms preserving a volume form (see [ACW], [T1], [Wi]). • The extension of the ergodic theory [Bo] of hyperbolic systems (as the existence of SRB or physical measures) to a wider class it has been the interest of researchers for a long time. Systems having dominated splitting are in the class where successful extensions have taken place (see for instance [BDV] and references therein). • Pesin Theory ([Pe]and[BaPe]) dramatically fail in the C1 topology: there are hyperbolic measures where the stable and unstable manifolds of any point in the support of the measure are trivial ([BCS]). Nevertheless, much of Pesin theory on hyperbolic measures can be recovered in the C1 category provided the Oseledet’s splitting is a dominated splitting, as claimed in [M6]. For instance in [ABC] it is proven the existence of stable and unstable manifolds and a version of Katok’s closing lemma [K] (an early adaptation in two dimensions appeared in [Ga2]); in [G] the Katok’s horseshoe construction ([K]) is extended to the C1 case; and Pesin’s entropy formula has been extended as well (see [CCE], [ST]and [T2]). • The entropy conjecture ([Sh2]) has been settled for diffeomorphism that are not approximated by ones having a homoclinic tangency [LVY]. Here the existence of dominated splitting is fundamental (see also [DFPV], [CSY]). • Likewise, dominated splitting has to do with flows, specially in the understanding of the Lorenz attractor and the theory of singular hyperbolic flows (see for instance [MPP]andthebook[AP]). A complete survey on the subject will result in a very long paper or just in a nonsense collection of results. To avoid both I tried to focus on some aspects and results to give the flavor of the subject. What is this survey about? It is about the dynamical consequences we may extract from a set with dominated splitting just using the structure itself (without assuming other hypothesis like being far from

152 MART´IN SAMBARINO homoclinic tangencies, heterodimensional cycles, or preserving a measure, etc). And so many important contributions (besides the ones mentioned in the above paragraphs) might not appear here. I apologize for this in advance. Acknowledgements: I wish to thank Sylvain Crovisier, Enrique Pujals and specially Rafael Potrie for reading a preliminary draft and for their comments, corrections and suggestions. And to the referee for the careful reading and important corrections and suggestions to improve the presentation of this survey.

2. Global Dynamics Given a dynamical system, i.e. a diffeomorphism (or homeomorphism) f : M → M, one would like to understand the asymptotic behavior of the orbits O(x)={f n(x):n ∈ Z} for any x ∈ M. Two dynamical systems f,g : M → M are said to be equivalent if there exists a homeomorphism h : M → M such that h ◦ f = g ◦ h. In other words, up to a continuous change of coordinates, the two systems f and g are the same. Can we describe or classify the dynamics of any system (up to equivalence)? Given a smooth manifold, let Diffr(M)={f : M → M,f is a Cr diffeomorphism} be the space of Cr diffeomorphisms endowed with the Cr topology. Can we describe or classify the dynamics of any f ∈ Diffr(M) (up to equivalence)? Under any reasonable sense of classification, this is an impossible task, since for instance any compact set of the two sphere S2 can be realized as the set of fixed points of a diffeomorphism f : S2 → S2 and so we are lead to classify the compact subsets of S2 up to homeomorphism. Poincaré was the first to understand that we would not try to describe all systems. We should avoid degeneracies or pathologies and attempt to describe the dynamics of the majority of the systems, and this is what global dynamics is about. What we understand by majority? There is no formal agreement, sometimes is con- 1 sidered open and dense or residual or Gδ-dense or just even dense. Through this paper generic stands for residual or Gδ-dense sets. After Thom’s Transversality Theorem, many results appeared in trying to describe the dynamics of most systems. Kupka-Smale Theorem is an important example of the above: There is a residual subset of Diffr(M) such that for any diffeomorphism in it, all its periodic points are hyperbolic and their stable and unstable manifolds intersect transversally (see Section 2.1 for the definitions). The notion of stability plays a central role in the theory: a dynamical system f : M → M is said to be (structurally) stable if all systems nearby are equivalent to it. Can we say which systems are stable? There is a famous conjecture by Palis-Smale, known as the stability conjecture, characterizing stability in terms of hyperbolicity. It is worth to mention the pioneering work of Peixoto, a landmark in this theory, which states that for Cr vector fields on surfaces, those who have finitely many critical elements (singularities and periodic orbits) and structurally stable (in the sense of vector fields) are Cr open and dense (these systems are known today as Morse-Smale dynamical systems). This lead Smale to naively conjecture the above was always the case. Nevertheless, previously works of Birkhoff [Bi], Cartwright- Litellwood [CaLi] and Levinson [L] showed that it was not the case and that more complex behavior (for instance having infinitely many periodic points) might not

1Other notion is: full Lebesgue measure set of parameters when we restrict to (generic) ﬁnitely parametrized families of diﬀeomorphisms.

DOMINATED SPLITTING 153 be destroyed by small perturbations. Also R. Thom suggested that hyperbolic linear automorphisms on the torus (a particular example of the so called nowadays Anosov diffeomorphisms) could exhibit this complex behavior as well. The above was the starting point for the creation of hyperbolic theory and the “dream” in the sixties was that generic systems were hyperbolic and stable. Nevertheless, open sets of non-hyperbolic (and unstable) diffeomorphisms were discovered and a new era appeared: the understanding of non-hyperbolic diffeomorphisms. Can we describe or characterize non-hyperbolic diffeomorphisms? What kind of bifurcations could appear? What kink of dynamic phenomena exists in the complement of hyperbolic dynamics? Still today this is an active research field. One would try also to describe the dynamics of large or big sets of systems. In this sense, the most natural is to consider open sets in Diffr(M). Besides stability one could consider other types of open set as well. For instance, given a dynamical property (P), one studies the systems such that the systems itself and all nearby ones share the property (P). In this case we say that the system has robustly the property (P) and the aim is to characterize those systems or understand the underlying mechanism that makes the property (P) to hold robustly. The general argument when trying to characterize a robust property (as stability) goes as follows: if the characterization we are looking for fails, then with an arbitrarily small perturbation we get a contradiction. However, in order to this argument work, we have to keep dynamic information after the perturbation (or in other words, to keep controlling of the dynamics while perturbing the system). These kind of perturbation techniques are known just in the C1 topology and this is the reason why most results in global dynamics are known in this category. This is due to the fact that the C1 topology is essentially linear and perturbing a diffeomorphism at a small scale is like perturbing linear maps. Nevertheless, many of the results regarding C1 global dynamics should be understood in the following sense: a smooth diffeomorphism can be C1 approximated by another smooth diffeomorphism having some desired dynamical property.

2.1. Hyperbolic dynamics: a brief overview. Hyperbolic theory, mostly developed in the sixties, is the basis of global dynamics. It is the paradigm of chaotic yet stable systems, and not only have a very nice description from the topological point view but form the ergodic one as well. Its methods are present in the study of global dynamics. Lets recall the definition. Definition 2.1. Let f : M → M be a diffeomorphism of a compact riemannian manifold without boundary. A compact invariant set Λ ⊂ M is called hyperbolic s u provided the tangent bundle over Λ splits into two subbundles TΛM = E ⊕ E such that: • s s u u Invariance: DfxEx = Ef(x) and DfxEx = Ef(x) • There exist C>0and0<λ<1 such that n n −n n Df s ≤Cλ ,n≥ 0andDf u ≤Cλ ,n≥ 0. /Ex /Ex s u The subspaces Ex and Ex are called stable and unstable respectively and vary continuously with the base point x and hence their dimensions are locally constant. The nice description of the dynamical behavior for hyperbolic sets comes through the fact that the hyperbolic splitting is reflected on the manifold itself (at small scale) through the following theorem (see [HPS]):

154 MART´IN SAMBARINO

Dfx Eu(f(x))

Eu(x) Tf(x)M

Es(x) f(x) Es(f(x)) x

TxM

Figure 1. hyperbolicity

Theorem 2.1. [Stable and Unstable Manifold Theorem for Hyperbolic Sets] Let f : M → M be a Cr diﬀeomorphism and let Λ a hyperbolic set. Then, there exists >0 such that the sets s { ∈ n n ≥ } W (x)= y M : dist(f (x),f (y)) <,n 0 and u { ∈ n n ≤ } W (x)= y M : dist(f (x),f (y)) <,n 0 are Cr embedded submanifolds for any x ∈ Λ such that • s s u u TxW (x)=Ex and TxW (x)=Ex . • ∈ s n n → If y W (x) then dist(f (x),f (y)) n→+∞ 0 and exponentially fast. ∈ u n n → Similarly, if y W (x) then dist(f (x),f (y)) n→−∞ 0 and exponentially fast. In particular, s { ∈ n n → } ∪ −n s n W (x):= y M : dist(f (x),f (y)) n→+∞ 0 = n≥0f (W (f (x)) and it is a Cr injectively immersed submanifold. Similarly for W u(x).

s u s u The sets W (x)(resp.W (x)) and W (x)(resp.W (x)) are called the stable (resp. unstable) manifold and the local stable (resp. unstable) manifold respectively. Thus, on the manifold itself we have two transverse local submanifolds (the local stable and unstable ones) such that are respectively forward and backward invariant and points in the stable (resp. unstable) get closer exponentially fast in the future (resp. in the past). This structure is the ultimate reason for the understanding of the dynamics of a hyperbolic set. Let us mention some main examples of hyperbolic sets. Consider ﬁrst a periodic point p of a diﬀeomorphism f : M → M, i.e., there exists n such that f n(p)=p. We say that p is hyperbolic periodic point if O(p) is a hyperbolic set. This is n → equivalent to the fact that Dfp : TpM TpM has no eigenvalues of modulus one. The index of a periodic point is the dimension of its stable subspace (or the dimension of its stable manifold). Hyperbolic periodic points play a fundamental role in the study of the dynamics, particulary through the study of the intersection between the stable manifold and the unstable manifold (of the orbit) of p. When the stable and unstable of a periodic orbit intersects transversally at a point x, i.e.,

DOMINATED SPLITTING 155

f(Q)

W u(p)

W s(p)

p x Q

Figure 2. A transversal homoclinic point and the Smale’s horseshoe x ∈ W s(O(p)) !∩ W u(O(p)wesaythatx is a transversal homoclinic point (or orbit) associated to p. If q is another hyperbolic periodic point we say that p and q are homoclinically related if W s(O(p)) !∩ W u(O(q) = ∅ and W s(O(q)) !∩ W u(O(p) = ∅. The Birkhoff-Smale Theorem says that if x is a transversal homoclinic orbit associated to a hyperbolic periodic point then both x and p belong to a (locally maximal)2 hyperbolic set and in particular this implies that x is accumulated by other periodic points. The homoclinic class H(O(p)) of a hyperbolic periodic point p is the closure of the set of transversal intersections between W s(O(p)) and W u(O(p)). The homoclinic class H(O(p)) coincides with the closure of the set of hyperbolic periodic orbits homoclinically related to it. One should be careful: despite the Birkhoff-Smale Theorem, in general the homoclinic class H(O(p)) is not a hyperbolic set. If p is a hyperbolic orbit then there exits a neighborhood U(f)wherep has continuation: for any g ∈U(f) there exists a unique g-periodic orbit pg contained in a neighborhood of the orbit of p. Thus, it is natural to consider the continuation of the homoclinic class H(O(pg),g). However, the term “continuation” here could be misleading: if H(p, f) is not hyperbolic or it does not coincide with the chain- recurrent class of p, the homoclinic class H(O(pg),g) could “explode” (see [Pa2]), although it can never “implode”. Smale’s horseshoe is a fundamental example in dynamical systems. It is a geometric description of the action of a planar diffeomorphism f on a square Q as depicted in the figure. One can prove that the maximal invariant set in the square n Q, Λ=∩n∈Zf (Q), i.e., the set of points whose orbit remains in Q, is a Cantor set, hyperbolic for f (which is in fact the homoclinic class of any of the two fixed points in Q) and can be modeled as the backward shift of two symbols, i.e., the dynamics of f in Λ is equivalent to σ :Σ→ ΣwhereΣ={0, 1}Z and σ(x)(n)=x(n +1). Another main example of a hyperbolic set is when the whole manifold M itself is a hyperbolic set. In this case the diffeomorphism f is called an Anosov diffeomorphism. A famous and particular example is the hyperbolic toral automorphism

2We say that a compact invariant set Λ is locally maximal if there is a neighborhood U such n that Λ = ∩n∈Zf (U)

156 MARTÍN SAMBARINO " # 21 on T2 induced by the (hyperbolic) matrix A = . Anosov diffeomorphism 11 have a very rich structure, the stable and unstable manifolds at every point lead to two transversal foliations (stable and unstable). D.V Anosov [A]provedthat Anosov diffeomorphism are structurally stable and, provided they are C2 and preserve the Lebesgue measure, they are ergodic (i.e. every invariant set has measure zero o full measure). Beautiful works by Franks [F1] and Manning [M] attempt to the classification of Anosov diffeomorphisms and which manifolds supports them. A fundamental question regarding Anosov diffeomorphisms is still open: Are Anosov diffeomorphism always transitive (i.e., have a dense orbit)? Hyperbolic theory focus the attention to those diffeomorphisms where the relevant part of the dynamics is a hyperbolic set, roughly speaking, where the recurrence takes part. Let us give two definitions. Given f : M → M we say that x is a wandering point if there exists a neighborhood U of x such that f n(U) ∩ U = ∅ for any n ≥ 1, otherwisewesaythatx is non-wandering. The set of non-wandering points is called the non-wandering set of f and it is denoted by Ω(f). Another important notion is the chain recurrent set. An -chain form x to y is a finite sequence x0 = x, x1,...,xn = y such that dist(f(xi),xi+1) <,for i =0,...,n− 1. The chain recurrent set R(f)isthesetofpointsx ∈ M such that there exists an -chain from x to itself for any >0. Inside the chain-recurrent set R(f) we have an equivalence relation: x ∼ y if we can go from x to y and from y to x with -chains for any >0. An equivalent class for this relation is called a chain-recurrent class.Chain recurrent classes are compact, invariant and pairwise disjoint. More generally, a set Λ ⊂R(f) is chain-transitive if for any x, y ∈ Λthereisan-chain from x to y. A chain-recurrent class is a chain-transitive set which is maximal for the inclusion. It always hold that periodic points are non-wandering and that Ω(f) ⊂R(f). If p is a hyperbolic periodic point then its homoclinic class is always contained in its chain recurrent class. We say that a diffeomorphism f : M → M is R(f)-hyperbolic (or just hyperbolic) if R(f) is hyperbolic set. A diffeomorphism is called Axiom A (unfortunately this awkward name has stuck into the literature) if Ω(f) is hyperbolic and the periodic points are dense in Ω(f). It holds that if f : M → M is R(f)-hyperbolic then it is Axiom A3. An Anosov diffeomorphism is R(f)-hyperbolic as well. The main dynamical description of a hyperbolic diffeomorphism is the following theorem: Theorem 2.2. [Spectral Decomposition Theorem [S1]]Letf : M → M be a R(f)-hyperbolic diffeomorphism. Then, there are finitely many chain recurrent classes, R(f)=Λ1 ∪ ...∪ Λk, each chain recurrent class Λi is transitive and it is the homoclinic class of a periodic orbit. In this case, the dynamics on each chain recurrent class (also called a basic piece) can be modeled by subshifts of finite type and are a classical object of thermodynamic formalism (see [Bo]). Besides its nice dynamic description, hyperbolic diffeomorphisms have the fundamental property of stability.ACr diffeomorphism f : M → M is called Cr structurally stable if there is a neighborhood U(f) ⊂ Diffr(M) such that for any g ∈Uthere is a homeomorphism h : M → M such that h◦f = g◦h. Weaker forms of

3Indeed, R(f)-hyperbolic is equivalent to Axiom is A and the no cycle condition. For the sake of simplicity we are not going to deﬁne this condition

DOMINATED SPLITTING 157 stability were introduced once it was known that structural stability was not generic ([S2]), the most important is known as Ω-stability. We say that f : M → M is Cr Ω-stable if there exits is a neighborhood U(f) ⊂ Diffr(M) such that for any g ∈U there is a homeomorphism h :Ω(f) → Ω(g) such that h ◦ f/Ω(f) = g/Ω(g) ◦ h. The Stability Conjecture by Palis-Smale [PaSm] states that f is Cr structurally stable if and only if f is R-hyperbolic and stable and unstable manifolds intersects transversally. The sufficiency was proven by Robin [Ro] and Robinson [R2]. The necessity was prove by Mañé[M1]intheC1 category. Although it was not included in the paper by Palis-Smale, it is very natural to extend the conjecture for Ω-stability (see [PaT]): a diffeomorphism f is Cr Ω-stable if and only if f is R-hyperbolic. By the works of Smale, Palis, and Mañé one has: Theorem 2.3. Let f : M → M be a C1 diffeomorphism. Then, f is C1 Ω-stable if and only if f is R-hyperbolic.

We end this section with the deﬁnition of Dominated Splitting to compare it with the one of hyperbolicity: Definition 2.2. Let Λ be an invariant set of f : M → M. We say that Λ has a dominated splitting provided the tangent bundle over Λ splits into two subbundles TΛM = E ⊕ F such that (i) E and F are invariant by Df. (ii) The subbundles E and F are continuous, i.e., Ex and Fx vary continuously with x ∈ Λ. (iii) There exist C>0and0<λ<1 such that for any x ∈ Λ Dfn Df−n ≤Cλn,n≥ 0. /Ex Ffn(x) The main diﬀerence with hyperbolicity is that we don’t know aprioriwhether E or F has a uniform behavior (contracting or expanding). And of course there are examples where they do not have a uniform behavior! Nevertheless, for the understanding of the dynamical consequences of having dominated splitting much of the ideas and methods come from hyperbolic theory. We will study some of this dynamical consequences along this paper. A (locally invariant) plaque family tangent to E is a continuous map D from the linear bundle E over Λ into M satisfying: 1 –Foreachx ∈ Λ the induced map Dx : Ex → M is a C embedding that is tangent to Ex at the point Dx(0) = x. 1 – The family (Dx)x∈Λ of C -embedding is continuous. – The plaque family is locally forward invariant, i.e., there exists a neighborhood U of the section 0 in E such that for each x ∈ Λ, the image of Dx(Ex ∩ U)byf is contained in Df(x)(Ef(x)). From [HPS] we know that there exist locally invariant plaque families tangent to E and F although we don’t know if they have dynamic properties (compare with the Stable* Manifold Theorem 2.1). However, if we know that for some γ<1we have that n Df ≤γn for any n ≥ 0 then the local stable manifold at x i=0 /Efi(x) has a uniform size (depends only on γ) and coincides with (a neighborhood) of the plaque (tangent to E)atx.

158 MART´IN SAMBARINO

2.2. Obstruction to hyperbolicity. Theorems 2.2 and 2.3 resumes the main features of hyperbolic theory: its beautiful dynamic description and characterization in terms of stability. Nevertheless, hyperbolicity was less universal than it was originally thought and open sets of non-hyperbolic diffeomorphisms were discovered. At the end of the sixties Abraham and Smale ([AS]) gave such an example. The idea is to construct two hyperbolic sets of different indices (i.e. different dimension of the stable space) but in the same chain recurrent class. For instance, consider a diffeomorphism f having a heterodimensional cycle: two fixed (periodic) points P, Q of different indices (see Figure 3), such that W s(P )∩ W u(Q) = ∅ and W u(P )∩W s(Q) = ∅. It is not difficult to see that P and Q belongs to the same chain-recurrent class. In this case f can not be R-hyperbolic. Is this robust? A priori is not, since at least one of the above intersections must be non- transversal due to the lack of enough dimensions. The key point in the Abraham- Smale construction is that a hyperbolic set Λ could have index i and the stable manifold of Λ (W s(Λ) = {y : dist(f n(y), Λ) → 0})beofdimensioni +1. Consider f : T2 → T2 an Anosov diffeomorphism and g : S2 → S2 having a hyperbolic fixed point p of index 1. Then Λ = T2 ×{p} is a hyperbolic set for f × g having index two. However W s(Λ) = T2 × W s(p) has dimension three.

q p

W s(q)

W u(p)

Figure 3. Heterodimensional cycle

And now it is not difficult how to construct an example of open sets of non- hyperbolic diffeomorphisms. In Figure 4 we have f : M → M with dimM =4 T2 ⊂ u s and a torus Λ = M where f/Λ1 is Anosov and dimW (Λ) = dimW (Λ) = 3 and a fixed point p of index one and the intersection of W u(p)andW s(Λ) is transversal as well it is the intersection between W s(p)andW u(Λ). Thus, p and Λ belongs to the same chain-recurrent class and so f is non-hyperbolic. Moreover, this structure is preserved under C1 perturbations. Indeed this is an example of a robust heterodimensional cycle: the existence of two hyperbolic sets of different indices Λ and Σ for f having continuations Λg and Σg for g in a neighborhood U(f) u s s u and such that W (Λg) ∩ W (Σg) = ∅ and W (Λg) ∩ W (Σg) = ∅ for any g ∈U. This idea was exploited by Bonatti and Diaz in the creation of blenders in an impressive sequence of papers and results ([BD1], [BD2], [BD3], [BD4]): a

DOMINATED SPLITTING 159

W s(p)

u W u(T2) W (p)

T2 W s(T2)

Figure 4 hyperbolic set such that the stable manifold of the hyperbolic set looks like it has bigger dimension than the stable manifold of each point of the set. With this construction, they gave new examples of robustly transitive diffeomorphisms, constructed robust heterodimensional cycles, proved the existence of Newhouse phenomena in the C1 category and the existence of wild and universal dynamics (see below). And they showed as well that a heterodimensional cycle is not an isolated phenomena: given f having a heterodimensional cycle associated to two periodic points p, q such that index(p) − index(q)=1, then there is g arbitrarily close to f having a robust heterodimensional cycle. The examples of open sets on non-hyperbolic diffeomorphism constructed at the end of the sixties do not include the case of surfaces since, as we have seen, they involve the presence of a heterodimensional cycle, a feature that can not exist if the manifold has dimension two. It was Newhouse ([N1], [N2], [N3]), through a truly original and remarkable idea, who constructed open sets of non-hyperbolic diffeomorphisms on surfaces. We say that f : M → M has a homoclinic tangency if there exists a hyperbolic periodic point p such that W s(p)andW u(p) have a non-transverse intersection. In this case, f can not be hyperbolic, since the point in this intersection and p belongs to the same chain-recurrent class and so this class can not be a hyperbolic set. Newhouse was able to construct robust tangencies: a hyperbolic set Λ for a 2 surface diffeomorphism f having a continuation Λg for g in a C neighborhood s u U(f) such that W (Λg)andW (Λg) have a non-transverse intersection outside a r Λg. (this construction is known in the C category, r ≥ 2 and still unknown in the C1 topology). Besides, Newhouse proved that this led to a surprising phenomena: the existence of a residual subset G⊂U(f) such that any g ∈Gdisplays infinitely many periodic attractors o repellers (sinks or sources). Nowadays this is known as New- house Phenomena. Furthermore, he proved that whenever a diffeomorphism exhibit a homoclinic tangency, then nearby there exists robust tangencies and Newhouse phenomena. In particular, a diffeomorphism having infinitely many periodic attractors or repellers have infinitely many chain recurrent classes! (this was not the

160 MART´IN SAMBARINO

W u(p)

W s(p)

Figure 5. Homoclinic tangency case in Abraham-Smale example). Thus, we have found residual subsets of open set of systems having infinitely many chain-recurrent classes. As mentioned before, Bonatti and Diaz ([BD1]) were able to show the existence of C1 Newhouse phenomena in manifolds whose dimension is at least three. This is done through the construction of a robust heterodimensional cycle involving two periodic points of different indices and with complex eigenvalues. The case of surfaces within the C1 topology remains open: Problem 1. Are there open sets in Diff1(M 2) formed by non-hyperbolic diffeomorphisms? We have seen that heterodimensional cycles and homoclinic tangencies are two main features to prevent hyperbolicity. In the eighties, Palis ([PaT], [Pa1]) conjectured that these are the main mechanism to understand non-hyperbolic diffeomorphisms: Conjecture [Palis] Any diffeomorphism in Diffr(M) can be Cr-approximated by one satisfying one of the following: • It is a hyperbolic diffeomorphism • It has a heterodimensional cycle • It has a homoclinic tangency. This conjecture was proven to be true in Diff1(M 2)([PS2]) and major advances where done for manifolds with dimension greater than two by Crovisier and Pujals ([CP]), still in the C1-topology. The Cr case with r ≥ 2 seems to be beyond reach in the next years. We have shown that there are residual subsets of open sets of diffeomorphisms displaying infinitely many chain-recurrence classes. A natural problem follows: Problem 2. Characterize those diffeomorphisms having (robustly) finitely many chain-recurrence classes (these diffeomorphisms are called Tame). In this direction, Bonatti [B] conjectured that a sufficient condition is to be far from homoclinic tangencies: Conjecture [Bonatti] Assume that f ∈ Diff1(M) can not be C1-approximated by one exhibiting a homoclinic tangency. Then f is Tame.

DOMINATED SPLITTING 161

2.3. Homoclinic tangencies versus dominated splitting. We have mentioned before that the presence of a homoclinic tangency prevents the diﬀeomor- phism to be hyperbolic. Indeed, it prevents the existence of a dominated splitting in the chain recurrence class of the periodic point associated to the tangency and of the same dimension as the index of the periodic point. Let’s be more precise. Let p be a hyperbolic periodic point an let x be a point of non-transverse intersection between W s(p)andW u(p). Assume that we have a dominated splitting s TΛM = E ⊕ F where Λ is the chain recurrence class of p and dimE = dimW (p). s u It must hold that E(p)=Ep and F (p)=Ep , that is, at the periodic point p the subspaces E and F must coincide with the hyperbolic splitting of the periodic s u point. On the other hand, at the point x, since TxW (p)∩TxW (p) = {0}, we have s u either E TxW (p)orF TxW (p). In both cases we arrive to a contradiction, iterating forward in the former or backwards in the latter (we violate the continuity of the splitting at p using E(f n(x)) or F (f −n(x))). More important is that this holds the other way around: the non-existence of dominated splitting yields by C1 arbitrarily small perturbation to the existence of a homoclinic tangency. This was proved in [PS2] in the case of surfaces and by Wen [We1] in any dimension, and represented a fundamental step towards the Palis Conjecture. Later Gourmelon [Go2] gave a local version (inside the homoclinic class) of the above:

Theorem 2.4. Let f : M → M be a C1 diﬀeomorphism and let p be a hyperbolic periodic orbit whose homoclinic class is not trivial. If the homoclinic class H(p, f) does not admit a dominated splitting of the same dimension as the index of p then there is a C1 arbitrarily small perturbation of f having a homoclinic tangency associated to (the continuation of) p.

2.4. Lack of domination and universal dynamics. The complete absence of dominated splitting yields strange and pathological behavior. The key point is the following: assume that f has infinitely many periodic orbits with unbounded periods and assume that this set has no dominated splitting at all (of any dimension), then by arbitrarily C1 small perturbation one have a diffeomorphisms having a periodic orbit such that the derivative at the period is a homothety. This was obtained in different contexts by [DPU], [BDP], [BGV], [BB]. In particular, through the construction of a robust heterodimensional cycle in dimension three involving two fixed points p, q of different indices (say index(p)= 2,index(q) = 1) such that p has a stable complex eigenvalue and q has an unstable complex eigenvalue (this forbid any domination in the heterodimensional cycle), Bonatti and Diaz [BD2] proved the following:

Theorem 2.5. ThereexistsanopensetW⊂Diff1(M) such that there exists a residual subset U of W such that for any f ∈U,anyopensetV of the homoclinic O⊂ 1,+ D3 1 class H(p, f) and any open set Diffint ( ) (the space of C diﬀeomorphisms orientation preserving of the closed ball into its interior) there is a periodic ball ⊂ k ∈O D V such that f/D is smoothly conjugate to some g . This implies in particular that the homoclinic class H(p, f)iswild(itisaccu- mulated by other chain-recurrent classes) and gives the Newhouse phenomena as well.

162 MART´IN SAMBARINO

2.5. Perturbation lemmas in the C1 topology. Periodic points play a fundamental role in dynamics from many different viewpoints. One fundamental problem solved by Pugh [Pu1] and known as Pugh’s Closing Lemma states that if x is a nonwandering point of f : M → M one can perturb f in the C1 topology to obtain g so that x is a periodic point for g (the Cr closing lemma with r ≥ 2isnot known). As a consequence of Pugh’s closing lemma one get that C1 generically the closure of the set of periodic points is equal to the nonwandering set, Per(f)=Ω(f). However, from the proof one can not conclude that the g-orbit of x shadows the f-orbit of x. This difficulty was solved in [M5]. A probability measure μ is invariant for f if μ(f −1(A)) = μ(A) for any Borel set A. We say that μ is ergodic if every invariant set has measure zero or one. From Birkhoff ergodic theorem we know that if μ is ergodic then for μ a.e. x we have that the sequence of measure equally distributed along pieces of orbits 1 k−1 μk = k i=0 δf i(x) converge to μ. Theorem 2.6 (Mañé’s Ergodic Closing Lemma). Let f ∈ Diff1(M) and let μ be an invariant probability measure. Then, for any neighborhood U(f) ⊂ Diff1(M) and >0,forμ − a.e.x ∈ M there exists g ∈U(f) and a g-periodic point p such that dist(gj (p),fj(x)) < j=0, 1,...,n(p) where n(p) is the period of p. Moreover, there exists a residual set D⊂Diff1(M) such that if f ∈Dand μ is an ergodic probability measure invariant under f there exists a sequence of f- periodic points p with periods n(p ) such that the measures μ = n n pn 1 n(pn)−1 δ j converges to μ in the weak topology. n(pn) j=0 f (pn) The second part appeared in [M6]. For a surprisingly simple and elegant proof of the above theorem see [C1]. Another fundamental perturbation techniques in the C1 topology extending Pugh’s closing lemma is the so-called Hayashi’s connecting lemma [H] and the extension to pseudo-orbits by Bonatti and Crovisier [BC]. We won’t state them since we won’t use it explicitly. Nevertheless lets mention some important consequences to have in mind. It holds C1 generically that: (a) Per(f)=Ω(f)=R(f); (b) the chain-recurrent class of a periodic point coincides with its homoclinic class; (c) the homoclinic classes of different periodic points are disjoint or coincide (in particular if two periodic points p, q of different indices belong to a robust heterodimensional cycle then generically H(p, f)=H(q, f). The next lemma, known as Frank’s lemma [F2]isasimpleyetpowerfultech- nique in the C1 topology. It says that if we choose linear maps close to the differential of f along a periodic orbit, these linear maps can be realized as the differential of a diffeomorphism g close to f keeping the orbit of the periodic point unchanged and the perturbation has arbitrarily small support. Lemma 2.1. Let M be a closed n-manifold and f : M → M be a C1 diffeomorphism, and let U(f) a neighborhood of f.Then,thereexistU1(f) ⊂U(f) and >0 such that if g ∈U1(f), S ⊂ M is a finite set, S = {p1,p2,...pm} and → − ≤ Li,i=1,...,m are linear maps Li : Tpi M Tf(pi)M satisfying Li Dpi g ∈U , i =1,...,m then there exists g˜ (f) satisfying g˜(pi)=g(pi) and Dpi g˜ = Li,i=1,...,m. Moreover, if U is any neighborhood of S then we may chose g˜ so that g˜(x)=g(x) for all x ∈{p1,p2 ...pm}∪(M\U).

DOMINATED SPLITTING 163

Let p be a periodic point of period k of f. The Lyapunov exponents are the 1 | | k numbers λ = k log σ for each eigenvalue σ of Dfp . We may consider as well the k Lyapunov exponents restricted to some invariant subspace of TpM under Dfp . If v i is an eigenvector, this is the same as log Df dμ where E i = Df () /Ex f (p) 1 k−1 and μ = k i=0 δf i(p). More generally, if μ is an ergodic and E is an invariant one dimensional subbundle, we deﬁne the Lyapunnov exponent along E as log Df/Ex dμ. We say that a Lyapunov is “weak” if it is close to zero. Franks’ Lemma allows us to change the index of a periodic if it has a weak Lyapunov exponent. This will be used in the following sections.

3. Dominated splitting: basic properties and examples The concept of dominated splitting was introduced by Mañéin[M2],[M3]. Apparently this notion already appeared in a letter from Mañé to J. Palis when he was an undergraduate student in Uruguay and before the celebrated conference in Salvador de Bahia in 1971. In this letter Mañé claimed he had a proof of the stability conjecture, something he would achieve almost twenty years later! Also, Pliss [Pl1]andLiao[L1] used this notion independently without naming it. And in [HPS] this notion is also hidden in the definition of eventually relatively normal hyperbolicity. Lets recall the definition we gave before: Let Λ be an invariant set of f : M → M. We say that Λ has dominated splitting provided the tangent bundle over Λ splits into two subbundles TΛM = E ⊕ F such that (i) E and F are invariant by Df. (ii) The subbundles E and F are continuous, i.e., Ex and Fx vary continuously with x ∈ Λ. (iii) There exist C>0and0<λ<1 such that for any x ∈ Λ (1) Dfn Df−n ≤Cλn,n≥ 0. /Ex Ffn(x) A way to understand the above definition is that any direction not contained in the subbundle E converges exponentially fast to the direction F under iteration of Df. Condition (1) is equivalent to the following: 1 (2) There exits m>0 such that Dfm Df−m ≤ /Ex Ffm(x) 2 and can be written also in the following form: m m Dfx vE 1 Dfx vF (3) For any vE ∈ Ex −{0} and VF ∈ Fx −{0} : ≤ . vE 2 vF Also, recalling that the co-norm or mininorm of a linear map A is m(A)=A−1−1 the above can be written as 1 (4) Dfm ≤ m(Dfm ) /Ex 2 /Fx The above suggest the terminology: the bundle E dominates the bundle F, and sometimesitiswrittenasE ≺ F. Condition (1) and the equivalent ones also imply that to “see” the domination we might have to wait some iterations. Notice that the dominated splitting does not depends on the riemannian metric, that is, a set having dominated splitting still it has it no matter if we change the

164 MARTÍN SAMBARINO metric on the manifold. Nevertheless, the constants C and λ above do depend on the metric. We can always find a riemannian metric on the manifold so that the constant C above is equal to 1, in other words we can “see” the domination in the first step. This is a result by N. Gourmelon [Go1]. Let’s see now an elementary property:

Proposition 3.1. Assume that Λ has a dominated splitting TΛM = E ⊕ F. Then this dominated splitting can be extended to Λ (the closure of Λ). Also, condition (ii) is equivalent to the following: the maps x → dim(Ex) and x → dim(Fx) for x ∈ Λ are locally constant. The angle between the subspaces Ex and Fx is bounded away from zero. Sketch of proof. → Let xn x be a sequence in Λ such that Exn and Fxn ˜ ˜ ˜ ˜ converge to subspaces Ex and Fx. By deﬁning Ef n(x) = Dfx(Ex) and similarly ˜ ˜ ˜ ˜ ˜ Ff n(x) we have that E and F satisfy (1). In particular TxM = Ex ⊕ Fx. Let yn be → also a sequence in Λ such that yn x and that Eyn and Fyn converges to subspaces denoted by Ex and Fx. Notice that dimEx = dimE˜x and dimFx = dimF˜x. We would like to show that Ex = E˜x. Otherwise, let v ∈ E,˜ v =1andv = vE + vF with vF =0 . Then n ≥ n ≥ n − n Df/E˜(x) Df v Df vF Df vE ≥ m(Dfn )v −Dfn v /F (x) F E(x) E m(Dfn ) = Dfn /F (x) v −v E(x) Dfn F E " E(x) # v ≥Dfn F −v E(x) Cλn E Dfn /E˜ →∞ ˜ and so n . Interchanging the roles of Ex and Ex we get a contradiction. Df/E From this one conclude that the dominated splitting can be extended to the closure of Λ and the equivalent condition stated in the proposition. No matter which reasonable deﬁnition of angle between subspaces we have, it is obvious that it is bounded away from zero. A set may have many dominated splittings. Let’s say that Λ has a dominated splitting TΛM = E⊕F of index i if dimEx = i for any x ∈ Λ. The above Proposition 3.1 also shows that a dominated splitting of index i on a set Λ is unique. Definition 3.1. Let Λ be an invariant set of f : M → M. Assume that we have a decomposition TΛM = E1 ⊕ E2 ⊕ ...⊕ Ek invariant under Df. We say that it is a dominated splitting provided

TΛM =(E1 ⊕ ...⊕ Ej) ⊕ (Ej+1 ⊕ ...⊕ Ek) is a dominated splitting for any j =1,...,k− 1. When the extremal subbundles E1 and Ek are uniformly contracting and expanding we call it partially hyperbolic (sometimes it is required that just one of the extremal bundles has a uniform behavior). Thus, on an invariant set Λ with dominated splittings we can consider the ﬁnest dominated splitting as the dominated splitting TΛM = E1 ⊕ E2 ⊕ ...⊕ Ek such that no Ei can be decomposed again so that the whole decomposition is dominated

DOMINATED SPLITTING 165 as well. This notion was introduced in [BDP]. The finest dominated splitting is unique. See Appendix B of [BDV]. Now we may ask: given the finest dominated splitting on a set, can we describe the dynamics? The question is twofold: on one hand what are the dynamical implications of such dominated splitting? and on the other one, which are the dynamical phenomena that prevents having a finer dominated splitting? These are very hard questions and we will try to shed some light on them in the following sections. Before we see some examples let’s characterize the domination in terms of cone fields4. A characterization of domination in terms of cones also appeared in [N4]. For x ∈ M and a>0, an a-cone of dimension n − i is a subset Ca(x)ofTxM such that we may find a direct decomposition TxM = E˜⊕F˜ with dimE˜ = i, dimF˜ = n−i such that C { ∈ ≤ } a = v TxM : v = vE˜ + vF˜ such that vE˜ a vF˜ . Proposition 3.2. Let Λ be an invariant set of f : M → M. Then Λ has a dominated splitting of index i if and only if there exist a map a :Λ→ R+ bounded away from zero and infinity, a cone field Ca(x)(x) of dimension n − i,anumber 0 <λ<1 and a positive integer n0 such that n0 C ⊂C Dfx ( a(x)) λa(f n0 (x)). Sketch of Proof. The direct implication follows immediately by the definition of a dominated splitting. For the converse, lets assume for simplicity that n =1. Define 0 −n n Ex = Df (Ca(f n(x))andFx = Df (Ca(f −n(x))). n≥0 n≥0

It can be proved that Ex and Fx are subspaces of dimension i and n−i, and invariant under Df. Now consider a tiny cone C(x) with respect to TxM = Ex ⊕ Fx. Then m C ⊂C m there exists m>0 such that Dfx ( a(x)) (f (x)) and from this one gets the domination. From Proposition 3.2 one easily gets the following important property of dominated splitting: it can be extended to a neighborhood and can not be destroyed by perturbations. Proposition 3.3. Let f ∈ Diff1(M) and let Λ be a compact invariant set having dominated splitting. Then there exist a compact neighborhood U(Λ) of Λ 1 and a neighborhood U(f) ⊂ Diff (M) of f such that for any g ∈U(f) the compact set gn(U) has dominated splitting. n∈Z Letsseesomesimpleexamplestohaveinmind: (1) A hyperbolic set is always a set with dominated splitting with E = Es and F = Eu (although it may have other dominated splittings). (2) Let p be a ﬁxed (periodic) point of f : M → M. Assume Dfp has all eigenvalues of moduli diﬀerent from σ>0 (and having eigenvalues greater and smaller than σ). Then p has a dominated splitting.

4For a more subtle characterization in terms of cones and a spectral gap see [BoG](and [Wj] as well).

166 MART´IN SAMBARINO

(3) Consider R : S1 → S1 an irrational rotation and f : S2 → S2 anorth- south dynamics. Let F = R×f. Then Λ = S1 ×{North} has a dominated 1 2 u splitting TΛS × S = E ⊕ E . In this case Λ is normally expanding curve supporting an irrational rotation. (4) Consider R : S1 → S1 an irrational rotation and f : T2 → T2 and anosov diffeomorphism. Then f × g : T3 → T3 has a (partially hyperbolic) splitting T T3 = Es ⊕ Ec ⊕ Eu. (5) Mañé Derived from Anosov diffeomorphism on T3 [M4]: it is a diffeomorphism f : T3 → T3 such that T T3 = Es ⊕Ec ⊕Eu is a partially hyperbolic splitting. This diffeomorphism have periodic points of different indices, it is transitive and any C1 small perturbation is transitive as well. This example is obtained by modifying a linear Anosov map on T3 by forcing a fixed point going trough a pitch-fork bifurcation (see also [BV], [PS1], [BFSV] for other properties).

uu Eλu E

I c Eλc p E

p q2 p q1

ss Eλs E

Figure 6. Ma˜n´e’s Derived from Anosov

(6) Bonatti-Viana example on T4 [BV]: it is a diﬀeomorphism f : T4 → T4 having a dominated splitting T T4 = Ecs ⊕ Ecu (which is the ﬁnest dominated splitting). It is also obtained by modifying an Anosov on T4 with hyperbolic structure T T4 = Es ⊕ Eu both bidimensional (see [BV], [BuFi], [T1] for other properties as well). Lets end this section with a simple yet useful characterization of uniform contracting subbundle whose proof we left to the reader: Lemma 3.1. Let Λ be compact invariant set of f : M → M and let E be a continuous and invariant subbundle over Λ. The following conditions are equivalent: (1) E is uniformly contracting: there exist C>0 and 0 <λ<1 such that Dfn ≤Cλn for any x ∈ Λ and n ≥ 0. /Ex n (2) lim → ∞ Df =0for any x ∈ Λ. n + /Ex (3) There exists m>0 such that for any x ∈ Λ there exists 0 0thereexistsxm ∈ Λ such that Df ≥ for 0 ≤ j ≤ m. Taking a /Exm 2 convergent subsequence of {xm} we conclude:

DOMINATED SPLITTING 167

Corollary 3.1. If the bundle E is not uniformly contracting then there exists x ∈ Λ such that j&−1 1 j ≤Df ≤ Df/E ∀j ≥ 0. 2 /Ex fi(x) i=0 4. Dominated Splitting and periodic points In this section we present some general results and ideas on dominated splitting (related mostly to periodic points). One would like to understand why a dominated splitting fails to be hyperbolic. The main goal (or conjecture) is that, under generic conditions (mind the third and fourth examples of the previous section), this can only happen due to the presence of periodic points of a different index of the dominated splitting. Theorem 4.1. Let f ∈ Diff1(M) and let Λ be a compact invariant set having a dominated splitting TΛM = E ⊕ F of index i. Then, if this decomposition is not hyperbolic then for any neighborhood U of Λ and every neighborhood U⊂Diff1(M) of f there exist g ∈U andahyperbolicg-periodic point p in U with index different from i. Sketch of proof: If the dominated splitting is not hyperbolic, then F is not uniformly expanded or E is not uniformly contracted. Assume that E is not uniformly contracted. Then there exists a point x ∈ Λ such that Πn Df ≥ 1 j=0 /Efj (x) 2 ≥ for any n 0 (recall Corollary 3.1). Consider the sequence of probability mea- 1 n−1 sures μn = n j=0 δf j (x).Letμ be an accumulation point of these measures. ≥ Then log( Df/Ex )dμ 0. The Ergodic Decomposition Theorem tell us that for μ a.e.x there exist ergodic measures νx such that for any continuous map h we have hdμ = ( hdνx)dμ. Hence, there exists an ergodic measure ν such that ≥ log( Df/Ex )dν 0. Let z be ν generic point. Now, we may assume that U and U are sufficiently small so that any invariant set of g ∈U in U has dominated splitting of index i. Let n → 0 . By the Ergodic Closing Lemma there exist gn ∈U j j and pn a periodic point of gn such that dist(f (x),gn(pn)) <n,j =0, .., n(pn). If n is small then the gn-orbit of pn is in U. Then, − n(p&n) 1 ≥ − (5) Dgn/E j 1 γn gn(pn) j=0 for some γn → 0 (that it is chosen in advance). By the domination ones get − n(p&n) 1 − Dg 1 ≤λn(pn) n/F j g (p ) j=0 n n for some λ<1. Now,(5)impliesthatforlargen we have that Dgn(pn) has a weak Lyapunov n/Epn exponent or can be perturbed to have it (see [M5]). Using Franks’ Lemma 2.1 one can thus obtaing ñ ∈U such that pn isg ñ periodic and the index of pn is less than i. The above result says, roughly speaking, that if a dominated splitting of index i over a set Λ is not hyperbolic it is due to the presence of hyperbolic periodic points of index = i. A major problem is whether these periodic points of different indices

168 MARTÍN SAMBARINO are attached to Λ. Let us be more precise: assume that Λ is the homoclinic class of a hyperbolic periodic point p of index i, and so, it has a natural continuation (for g close to f consider Λ(g) to be the homoclinic class of the continuation pg of p). If the dominated splitting is not hyperbolic, can we perturb f so that Λ(g) has a periodic point of different index? It is possible to have counter examples of this (see Remark 6.1) and the right question should be: if the dominated splitting is not hyperbolic, can we perturb f so that either Λ(g)ishyperbolicorcontainsa point of different index? Another way to ask the same is assuming some generic conditions on f : Problem 3. Let f ∈ Diffr(M), (r ≥ 1) and assume that f is Cr generic. Let p be a hyperbolic periodic point of index i and assume that the homoclinic class H(p) has a dominated splitting of index i. If this dominated splitting is not hyperbolic, is it true that H(p) contains a periodic point of index = i? This problem has been attacked in various different situations (always in the C1 topology), and with sophisticated techniques (see for instance [C1], [C2], [CP]). A possible approach would be to show that the homoclinic class H(p) has “weak periodic points of index i”. Let’s see some results in this direction. Some previous results are needed. The first one is Pliss’ Lemma, a fundamental tool to play with dominated splitting5.Itsaysthatifweseeafteralargetimelapseagoodcontrac- tion along a subbundle then there are (many) intermediary (Pliss) times where we see contraction step by step (in the literature they are also called hyperbolic times). Theorem 4.2 (Pliss’ Lemma [Pl2]). Let f : M → M be a diffeomorphism. Then, given 0 <γ1 <γ2 < 1 there exists a positive integer N = N(γ1,γ2) > 0 and ∈ ⊂ d>0 such that if for some*x M and some subspace E0 TxM and denoting j N ≤ N ≤ Ej = Df E0 we have that j=0 Df/Ej γ1 then there exist 0 n0 dN such that n&−1 n Df ≤γ ∀ 1 ≤ n ≤ N − ni,i=0,...,k. /Eni+j 2 j=0 The next result is a version of the Anosov Closing Lemma. Let Λ be a compact invariant set having dominated splitting TΛM = E ⊕ F. Let 0 <η<1. For x ∈ Λ and a positive integer n we say that (x, f n(x)) is η-string provided k&−1 k&−1 ≤ j −1 ≤ j Df/E j η ,k=1,...,n and Df − η ,k=1,...,n f (x) /Ffn j (x) j=0 j=0 The meaning of a string is that between x and f n(x) the bundles E and F have a (uniform) hyperbolic behavior (see Figure 7). Theorem 4.3. Let f : M → M be a diffeomorphism and let Λ be a compact invariant set having dominated splitting TΛM = E ⊕ F and let 0 <η<1 be given. ni Then, given >0 there exists δ>0 such that if (xi,f ) are η-strings, i =1,...,k ni n such that dist(f (xi),xi+1) <δ,for i =1,...,k − 1 and dist(f k (xk),x1) <δ) n then there exists a periodic point p, with f (p)=p where n = n1 + ···+ nk and

n1+···+nj−1+ dist(f (p),f (xj)) < 1 ≤ j ≤ k, 0 ≤ ≤ nj .

5Another useful tool is Liao’s Selection Lemma [L3] but we won’t state it since we won’t use it here.

DOMINATED SPLITTING 169

When k = 1, an explicit proof was given by Liao [L2]. A proof of the generalized version was done by Gan [Ga1]. The following theorem is a (simplified) version of a result proved by Mañé[M1] as an important step for the stability conjecture (Theorem II.1). See also [BGY]. The proof we present here is different from the one of Mañéor[BGY], although it is based on the same ideas.

F f m1

m1 p = f n(p) F f (x1) x2 E

E x1 m1 m3 f (p) f (x3)

f m2 F m2 f (x2)

m +m f 1 2 (p) x 3 E

f m3

Figure 7

Theorem 4.4. Let p be a hyperbolic periodic point. Assume that the homoclinic s class H(p) of p has a dominated splitting TH(p)M = E ⊕ F of the same index as the index of p and where Es is uniformly contracted. Then, either the homoclinic class is hyperbolic or there are periodic points in the class with arbitrarily weak Lyapunov exponent along F, or more precisely, given 0 <γ<1 there exists a periodic q ∈ H(p) such that n(&q)−1 n(q) −1 γ < Df − < 1 /Ff j (q) j=0 where n(q) is the period of q. Sketch of proof. Lets assume that H(p) is not hyperbolic. We may assume, ∈ changing the riemannian metric if necessary, that Df/Ex <λ<1 for any x H(p)(sinceEs is uniformly contracted). Moreover, from the fact p is hyperbolic and index(p)=dimEs we may also assume that Df−1 <λ.Fromthe /Ffj (p) fact that we have a dense set of periodic orbits homoclinically related to p that D spends most of its time near p we ﬁnd a dense* set in H(p) consisting of periodic ∈D n(q)−1 −1 n(q) orbits such that for any q we have Df − <λ1 for some j=0 /Ff j (q) λ<λ1 < 1. Now, ﬁx any γ such that λ1 <γ<1 and assume the homoclinic class

170 MARTÍN SAMBARINO is not hyperbolic. Choose γ<γ1 <γ2 < 1. From Pliss’ Lemma one gets that for every q ∈Dthere are points qi in the orbit of q such that n&−1 (6) Df−1 ≤γn for any n ≥ 1. /F −j 2 f (qi) j=0 ∈D For each q consider all the points q1,q2,...,qmq = q1 that satisfies (6) and ordered following the orbit of q. Notice that for any qi,qj both (qi,qj)and(qj,qi) are γ2-strings. Now lets look at the orbit distance between qi and qi+1,thatis mi mi, where f (qi)=qi+1. If mi is uniformly bounded for any i and for every q ∈Dwe conclude that H(p) is uniformly hyperbolic. Thus, we may find a se- D quence q(n)in and points qin (n),qin+1(n) (in the orbit of q(n)) whose orbit distance goes to infinity. And so, for n large enough, we get from Pliss Lemma that * − − mi 1 1 mi Df − >γ1 (otherwise we have a point satisfying (6) between j=0 /Ff j (q (n)) in+1 qin (n)andqin+1(n)) which is impossible since all the points satisfying this equation were in the list).

Now we may assume that qin (n)andqin+1(n) converge to points x and y.Let >0 be very small and let δ form the above closing lemma. Fix some n0 such that q (n )andq (n ) are at distance from x and y less than δ/2. And then, choose in0 0 in0+1 0 n much larger than n so that q (n )andq (n ) are at distance less than δ/2 1 0 in1 1 in1+1 1 of x and y. Notice that the γ strings (q (n ),q (n )) and (q (n ),q (n )) 2 in0+1 0 in0 0 in1 1 in1+1 1 are as in the Theorem 4.3 with k =2. Choose 0

mi min N C n0 ((1 − c)γ1) 1 >γ . * − Thus, if was small enough we conclude that γN < N 1 Df−1 . On the j=0 /F − * f j (z) n−1 −1 ≤ n ≤ ≤ other hand Df − ((1+c)γ2) for 1 n N. This implies that z has j=0 /Ff j (z) a uniform unstable manifold (see the end of Section 2.1) and therefore, and if and δ were chosen suﬃciently small, we have a heteroclinic intersection between z and q (n )andsoz belongs to the homoclinic class H(p). .Sinceγ was arbitrary we in0+1 0 deduce the existence of periodic points with arbitrarily weak Lyapunov exponent along F .

Very recently Xiadong Wang [Wa] proved a beautiful extension of the above theorem: Theorem 4.5. Let f ∈ Diff1(M) be C1 generic and let p be a hyperbolic periodic point of f. Assume that the homoclinic class H(p) has a dominated splitting TH(p)M = E ⊕ F with dimE ≤ index(p) and assume that the subbundle E is not uniformly contracting. Then, there are periodic points in H(p) having the maximal Lyapunov exponent along E arbitrarily close to 0. A natural problem therefore is: Problem 4. Let H(p) be a homoclinic class of a hyperbolic periodic point s ⊕ c ⊕···⊕ c ⊕ u s having a dominated splitting TH(p) = E E1 Ek E where E is uniformly u c contracted and E is uniformly expanded and Ei are one dimensional and neither

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c c E1 is uniformly contracted nor Ek is uniformly expanded. Do there exist periodic s points q1,q2 in H(p) such that the index of q1 is dimE and the index of q2 is dimEs +k? To avoid counterexamples (see the end of Section 6) one should assume that f satisﬁes some generic condition .

We remark that from Theorem 4.5 (and also by [CSY]) the indices dimEs +1 and dimEs + k − 1 are realized (and by [ABCDW] the indices between both are realized as well). The above problem admits a more general formulation (and more s c u diﬃcult?) when the class H(p) admits a dominated splitting TH(p) = E ⊕E ⊕E where Es and Eu are maximal (i.e., H(p) does not admit a dominated splitting E˜s ⊕ E˜c ⊕ E˜u with Es ⊂ E˜s and Eu ⊂ E˜u). A positive answer to the above problem will represent a fundamental step towards the Palis’ conjecture or Bonatti’s conjecture on ﬁniteness of chain-recurrent classes far from tangencies (see [B]).

5. Intermediate one dimensional subbundle Let f : M → M be a diﬀeomorphism and let Λ be a compact invariant set c c having a dominated splitting TΛ = E ⊕ E ⊕ F where E is one dimensional. Can we say something about the dynamics of Λ? Even in the case where E = Es is uniformly contracting and F = Eu is uniformly expanding this is a diﬃcult question.

Problem 5. Assume that f ∈ Diffr(M) is Cr generic and partially hyperbolic TM = Es ⊕ Ec ⊕ Eu with Ec one dimensional (and f is not Anosov). Is it true that the chain recurrent set R(f) has finitely many chain-recurrent classes? What s ⊕ c ⊕ ⊕ c ⊕ u c if TM = E E1 ... Ek E with Ei one dimensional? Lets explain some related problems as well. An Anosov flow (generated by s vector field X : M → TM) ϕt : M → M is a flow such that TM = E ⊕ u s u ⊕E where this splitting is invariant under Dϕt and the bundles E and E are uniformly contracted and expanded respectively. The time one map f = ϕ1 has a natural partially hyperbolic structure TM = Es ⊕ Ec ⊕ Eu where Ec is the bundle tangent to the orbits of the flow. Is it true that any Cr generic perturbation of f has finitely many chain-recurrent class? If the flow is not the suspension of an Anosov diffeomorphism, is any Cr perturbation transitive? There are two main reasons to consider dominated splitting with one-dimensional subbundle. On one hand this is the situation when we are far from homoclinic tangencies (see [PS2], [We1], [C2]). On the other one, although the action of Df along Ec might be neutral, since it is one dimensional, one could expect to have a description on the dynamics in the Ec direction. This is the content of a basic tool in the area known as Crovisier’s Center Models and developed in [C2]and[C3]. This tool was used to prove the major results on the Palis’ conjecture ([C3], [C2] and [CP]) and on the Bonatti’s conjecture ([B]) on finiteness of chain-recurrent classes far from tangencies as well (see [CSY]). Let us briefly explain this tool (we recommend to look at [C1] for this and many applications as well). By the theory in [HPS] one knows that there are a family of one dimensional manifolds Wc(x)forx ∈ Λ (center plaques) such that Wc c Wc Tx (x)=Ex and are locally invariant, that is, f( (x)) contains a neighborhood c c of f(x) within W (f(x)). This allows to lift the dynamics to TΛE .

172 MART´IN SAMBARINO

Definition 5.1. A center model (Λˆ, fˆ) associated to a compact invariant set c c Λandf with TΛ = E ⊕ E ⊕ F where dimE = 1 is a compact set Λˆ, a continuous map fˆ : Λˆ × [0, +∞)andamapπ : Λˆ × [0, +∞) → M such that: • fˆ(Λˆ ×{0})=Λˆ ×{0}. • fˆ is a local homeomorphism of a neighborhood of Λˆ ×{0}. ˆ ˆ ˆ ˆ • f can be written as a skew product f(x, t)=(f1(x), f2(x, t)). • π(Λˆ ×{0})=Λ. item π ◦ fˆ = f ◦ π. • The maps t → π(ˆx, t) forms a family of C1 embedding of [0, +∞)inM. • The curve π(ˆx × [0, +∞)) is tangent to Ec at π(ˆx, 0). One can consider two cases: the orientable case where there exists an orientation of Ec invariant by Df and where there is not. Theorem 5.1 ([C2],[C3]). Let Λ be a compact, invariant and chain-transitive c c set with dominated splitting TΛ = E ⊕ E ⊕ F where dimE =1. Then there exists a center model (Λˆ, fˆ) associated to (Λ,f) where Λˆ is chain-transitive. In the { }× ∞ orientable case, π/Λˆ×0 is a homeomorphism and π( xˆ [0, + )) is compatible c ˆ 1 c with the orientation on E , and in the non-orientable case Λ coincides with TΛE . Moreover, one of the following cases holds: • Type(R): (Chain-recurrent) For every >0 there exists a point x ∈ Λ and an arc γ,x ∈ γ ⊂Wc(x) such that the length of f n(γ) is bounded by andonecangofromγ to Λ and viceversa with arbitrarily small chains contained in a neighborhood of Λ. • Type(N): (Neutral) There exist a base of attracting open neighborhoods U of the zero section (i.e., fˆ(U) ⊂ U and a base of repelling open neighborhoods. • Type(H): (Hyperbolic) There exists a base of attracting (respect. repelling) open neighborhoods of the zero section and Wc is contained in the chain stable (respect. unstable) set of Λ. • Type(P): (Parabolic) In this case we are in the orientable case and on each “side” we have type (N) or (H). Thus, three subcases appear: PSU, c,+ c,− PNS,PNU depending on which cases appear on E and E . As we said before, using the central models one can get many interesting result. Just to give a rough idea let’s see the following: Theorem 5.2 ([C2],[C3]). Let f ∈ Diff1 be a C1 generic diffeomorphism and let Λ be a compact invariant chain-transitive set with dominated splitting TΛ = Es ⊕ Ec ⊕ Eu where Es is uniformly contracted, Eu is uniformly expanded and dimEc =1andofType(R).Then,Λ is contained in the chain-recurrent class of a periodic point. Sketch of Proof: Every point whose orbit remains in a neighborhood of Λ has a uniform (strong) stable Ws and unstable Wu manifolds (tangent to Es u s u and E ). Let γ be a chain recurrent segment. Then ∪c∈γ W (x)and∪c∈γ W (x) defines (topological) manifolds of dimension dimEs+1 and dimEu+1 and contained respectively in the chain stable ((i.e we can with -chains for arbitrarily small to Λ) and chain unstable set of Λ . Since f isgeneric(see[BC]and[C4]) one find a periodic orbit O whose orbit remains in a neighborhood of Λ and there is p ∈O close to the middle point of γ. Thus, the strong stable and unstable manifolds at p

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N R HPSU PSN PUN

Figure 8. Center Models intersects the chain unstable and chain stable set of Λ and so we can go with -chains form p to Λ and viceversa, implying that p and Λ are in the same chain-recurrent class.

The following result is a consequence of Theorem 1.2 of [CSY]:

Theorem 5.3. Let f be a C1 generic diﬀeomorphism and let Λ be a compact s⊕ c⊕ c ⊕ invariant chain transitive set having a dominated splitting TΛM = E E1 ...Ek u s u c E where E is uniformly contracted, E is uniformly expanded and dimEi =1,i= c c 1,...,k. And neither E1 is uniformly contracted nor Ek is uniformly expanded. Then one of the following holds: • c k =1and the Lyapunov exponent along E1 of every ergodic measure supported in Λ is zero. • Λ is contained in the chain-recurrent class of a periodic point.

c The idea is the following: if all Lyapunov exponents along E1 of every ergodic measure are zero then we are in the ﬁrst case. Otherwise there exists an ergodic measure μ whose Lyapunov exponent is non-zero and Theorem 1.2 of [CSY] applies and we are in the second case. Furthermore, it follows from the same theorem that, in the second case, Λ is contained in the local homoclinic class of a hyperbolic periodic orbit O in a neighborhood U of Λ (that is, the closure of the set of transversal intersections of the stable and unstable manifolds of O whose orbits are contained in U)andso s ⊕ c ⊕ c ⊕ u with the same structure E E1 ...Ek E . Problem 6. Let f be a Cr generic diﬀeomorphism. Does there exist an ape- riodic chain recurrent class C (i.e. a chain recurrent class without periodic points) s c u c partially hyperbolic TCM = E ⊕ E ⊕ E with dimE =1?

A negative answer to this question will represent a deﬁnite step towards the Palis’s conjecture or the Bonatti’s conjecture (see [B], [CP], [CSY]).

174 MART´IN SAMBARINO

6. Extremal one dimensional subbundle In this section we study compact invariant sets Λ having dominated splitting TΛM = E ⊕ F where E or F is one dimensional. We call this kind of splitting codimension one dominated splitting. Lets begin with the simplest case: when the ambient manifold M is compact surface (i.e. a bidimensional compact riemannian manifold). If Λ is a compact set having dominated splitting TΛM = E ⊕ F then both E and F are (extremal and) one dimensional. How such a set can fail to be hyperbolic? There are two trivial counterexamples: either Λ contains a non hyperbolic periodic point or Λ contains a periodic simple closed curve normally hyperbolic (and hence attracting or repelling) such that at the period the dynamics has irrational rotation number. The next result says that these are the only obstructions provided the diffeomorphism is at least of class C2. Theorem 6.1 ([PS2]). Let f ∈ Diff2(M) where M is bidimensional. Let Λ be a compact invariant set having a dominated splitting TΛM = E ⊕ F. Assume that any periodic point of f in Λ is hyperbolic and that Λ does not contain a periodic simple closed curve supporting an irrational rotation. Then Λ is hyperbolic. This result implies that if all periodic points are hyperbolic and no normally hyperbolic curve supporting an irrational rotation exists (which is a Cr generic condition for r ≥ 1) then a set having dominated splitting for f ∈ Diffr,r ≥ 2 is hyperbolic. And also implies that there exists a residual set D⊂Diff1(M) such that if f ∈Dand Λ is a compact invariant set having dominated splitting is hyperbolic. To prove this consider {Un} a countable basis of the topology of M. Consider the family {Vn} of finite collection of elements Un’s (which is countable). Let An the interior of the set 1 {f ∈ Diff (M): ifK ⊂Vn is compact, invariant having DS is hyperbolic} where DS stands for dominated splitting. Let Bn be the complement of the closure of An. Let Dn = An ∪Bn and D = ∩nDn. Let f ∈Dand let K a compact invariant set having dominated splitting. Let V be a compact neighborhood of K and let U be a neighborhood of f such that for any g ∈U the maximal invariant set of g in V has dominated splitting. Now, there exists Vn such that K ⊂Vn ⊂ V. Taking a 2 sequence gn of C generic diffeomorphism converging to f we see that the maximal invariant set of gn in V is hyperbolic and so f can not be in Bn. Therefore, f ∈An and so K is hyperbolic. So far there is no proof of this fact just using C1 techniques (i.e. without approximating by a C2 diffeomorphism and using Theorem 6.1). The proof of Theorem 6.1 is very technical. It is an extension of a result on non- critical one dimensional dynamics of Mañé([M7]). A very rough and general idea about the proof is: since both E and F are one dimensional extremal bundles, there Wcs Wcu exist locally invariant manifolds loc(x)and loc(x) tangent to E and F respec- ∈Wcs n n → tively having dynamical properties: if y loc(x)thendist(f (y),f (x)) n→∞ 0 Wcu and similarly for loc(x) in the past (in terms of Crovisier’s Center Models, they are respectively type(H)-attractive and type(H)-repelling). Then one prove that | n Wcs | ∞ | n Wcs | n≥0 f ( loc(x) < where f ( loc(x) denotes the length (the same for cu n W (x) in the past). Finally, the above implies that Df →→∞ 0and loc /Ex n −n 2 Df →n→∞ 0. For all these facts, the C assumption is crucial (in order to /Fx

DOMINATED SPLITTING 175 control distortion). Indeed, if f is just C1 the above theorem is false: at the end of this section we give a counterexample. Theorem 6.1 gives that dominated splitting on two dimensions of a C2 diffeomorphism imposes certain constraints. Thus, we may ask if we can fully describe the dynamics. The answer is yes (at least to some extent). Theorem 6.2 ([PS3]). Let f ∈ Diff2(M 2) and assume that the Limit Set L(f) has a dominated splitting. Then L(f) can be decomposed into L(f)=I∪ L˜(f) ∪R such that (1) I is a set of periodic points with bounded periods and contained in a disjoint union of finitely many normally hyperbolic periodic arcs or simple closed curves. (2) R is a finite union of normally hyperbolic periodic simple closed curves supporting an irrational rotation. (3) L˜(f) can be decomposed into a disjoint union of finitely many compact invariant and transitive sets. The periodic points are dense in L˜(f) and contains at most finitely many non-hyperbolic periodic points. The (basic) sets above are the union of finitely many (nontrivial) homoclinic classes. Furthermore f/L˜(f) is expansive. A fundamental step to prove the theorem above is the following rather surprising result: Theorem 6.3 ([PS3]). Let f : M → M be a C2-diffeomorphism of a two dimensional compact riemannian manifold M and let Λ be a compact invariant set having dominated splitting. Then, there exists an integer N1 > 0 such that any periodic point p ∈ Λ whose period is greater than N1, is a hyperbolic periodic point of saddle type. Now,letusturntodimM ≥ 3 and let Λ a compact invariant set having codimension one dominated splitting, say TΛM = E ⊕ F where F is one-dimensional. A natural extension of Theorem 6.1 should be: similar conditions as in Theorem 6.1 imply that F is uniformly expanding?. Some partial results were given in [PS4] (when E is uniformly contracted) and in [CP] with some condition on the topology of Λ. Nevertheless, it turns out to be true in general: Theorem 6.4 ([CPS]). Let f : M → M be a C2 diffeomorphism and let Λ be a compact invariant set having a dominated splitting TΛM = E ⊕ F with dimF =1. Assume that for any periodic points in Λ the Lyapunov exponent F is positive and that Λ does not contain a periodic simple closed curve tangent to F and supporting an irrational rotation. Then F is uniformly expanded. We finish this section giving a C1 counterexample for Theorem 6.1: Theorem 6.5. There exists a C1 diffeomorphism g : T2 → T2 having a dominated splitting T T2 = E ⊕ Eu such that Eu is uniformly expanded and: • Any periodic point of g is hyperbolic of saddle type • g is in the C1 closure of the set of Anosov diffeomorphisms. • g is conjugated to an Anosov diffeomorphism. • E is not uniformly contracted. For the (sketch of the) proof of this theorem we need some auxiliary lemmas. The first one is straightforward.

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Lemma 6.1. Given, σ>1 there exist η>0,β >0 and σ1 > 1 such that given h a local diffeomorphism around 0 in R2,h(0) = 0 and assuming that h can be written β as h(x, y)=(u(x, y),v(y)) and such that ux≤e , uy <ηand vy >σthen: • Cu { ∈ R2 ≤ } Cu ⊂Cu Denoting a = (u, v) : u a v we have Dh(x,y) 1 ρ where eβ +η ρ = σ < 1. • ∈Cu If w 1 then Dh(x,y)w >σ1. The next lemma will be the key to the induction argument we are going to do: Lemma 6.2. Let h(x, y)=(u(x, y),v(y)) be as the above Lemma. Let δ>0,α> β −λ 0,γ > 0,β > 0 and λ>−β be given with e < 2 and assume that 0 0 arbitrarily small and γ>0 there exist a function βk :[0, +∞) → R such that: ∞ − ≤ ≤ (1) βk is C , non-increasing and such that k βk(t)t 0. (2) βk is supported in [0.k], i.e. supp[βk] ⊂ [0,k]. (3) β(0) = eγ − 1 > 0. Define 2 g(x, y)=h(x, y)+(Z(y)β(x )ux(0, 0)x, 0). 2 Notice thatu ˜(x, y)=u(x, y)+Z(y)β(x )ux(0, 0)x. Let’s see that the conditions are fulfilled if k in the above sublemma was chosen small enough. It is immediately that g and h coincides outside a ball of radius δ at the origin. Notice that g(x, y)"− h(x, y)≤# β(0)eβk<αif k is small enough. Now, a a Dg = Dh + A with A = 1 2 and 00

2 2 2 2 a1 = Z(y)[β(x )ux(0, 0) + 2β (x )x ux(0, 0)] and a2 = Z (y)β(x )ux(0, 0)x. −λ −λ γ Then, a1≤β(0)e +2ke < 2β(0) if k is small. Observe that β(0) = e −1. 4 −λ On the other hand a2≤ β(0)e k<β(0) again if k is small enough. δ1 γ Moreoveru ˜x(0, 0) = ux(0, 0) + β(0)ux(0, 0) = e ux(0, 0). Finally, since ux > 0 and −k ≤ β (t)t ≤ 0wehavethat˜ux > 0ifk is small and −λ γ−λ u˜x≤ux + β(0)ux(0, 0)≤e (1 + β(0)) = e

Let’s continue with the proof of Theorem 6.5. The idea is to begin with a linear Anosov and then perform a sequence of perturbations (along periodic points) that converge in the C1 topology to the desired diﬀeomorphism.

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2 2 Start" with# the linear Anosov diﬀeomorphism f0 : T → T given by the matrix 21 A = . Let e−λ and eμ be the eigenvalues of A where λ and μ are positive 11 and let Es,Eu be the associated subspaces. Everything we do in local coordinates will be referred to the decomposition Es ⊕ Eu (both in the tangent space as in T2). μ Choose 1 <σ0,η > 0andσ1 from the ﬁrst lemma (and ρ as well). ≥ Choose a sequence n,n 1 such that n≥1 n <β.Now, set a sequence of positive numbers λn,n≥ 0 as follows: λ0 = λ and for n ≥ 1:

λ − λ − n 1 <λ < n 1 + . 2 n 2 n ˜ Let λn,n≥ 0 be a sequence such that

λ − λ − λ˜ = λ and n 1 <λ < λ˜ < n 1 + for n ≥ 1. 0 0 2 n n 2 n Set γ = λ˜ − λ ,n ≥ 0 and observe that γ <λ + < ∞ and that n n n+1 n≥0 n 0 n ˜ (λn − λn) < n <β. Now, let’s begin our induction process. Let p = p0 the fixed point of f0. Applying Lemma 6.2 around the fixed point p0 with λ = λ0 and γ = γ0 (δ and α 2 2 do not matter too much here), we get a diffeomorphism f1 : T → T where the s stable foliation (tangent to E )iskeptinvariant,p0 is fixed by f1,and s • The Lyapunov exponent of f1 at p0 is L (p0,f1)=−λ0 + γ0 = −λ1. That −λ1 is Df1/Es = e . γ0 • distC1 (f0,f1) ≤ 2(e − 1) . γ −λ λ β 2 •Df s ≤e 0 0 ≤ e 1 < 1

The true induction process starts here: pick a periodic point p1 of f1 with large period n1 that spends much of its time near p0 so that its (stable) Lyapunov s 1 ˜ ˜ ˜ exponent is L (p1)= log Df1/Es = −λ1 for some λ1 where λ1 < λ1 < n1 p1 λ0/2+1. j i − Now, pick δ>0 such that f1 (Bδ(f (p1))),j =0,...,n1 1 are disjoint for any − − O ∪n1 1 i i =0,...,n 1andthatp0 does not belong to Bδ( (p1)) = i=0 Bδ(f (p1)). Let j i α1 > 0 be such that if distC0 (f1,g) ≤ 2α1 then g (Bδ(f (p1))),j =0,...,n1 − 1 are disjoint for any i =0,...,n− 1. i i+1 Putting local coordinates at f (p1)andf (p1)fori =0,...,n1−1weperform 2 2 a perturbation using Lemma 6.2 with λ = λ1,γ = γ1 and we get f2 : T → T such that • j j ≥ f2 (p1)=f1 (p1),j 0. • The stable foliation (tangent to Es)iskeptinvariant. s s ˜ • L (p1,f2)=γ1 + L (p1,f1)= γ1 − λ1 = −λ2. γ +γ −λ β •Df s ≤e 1 0 0

178 MART´IN SAMBARINO

• f2 has dominated splitting (by Lemmas 6.1 and 6.2) with expanding di- Cu rection in the cone ρ • f2 is Anosov. The last point deserves a bit explanation (since at some point in the process γ +···+γ +γ −λ the inequality Df s ≤e n 1 0 0 does not ensure that Df s < n+1/Ex n+1/Ex 1 although is less than eβ enough to guarantee the domination). The general argument is as follows: since f1 is Anosov, we know that for some m>0wehave m Df s < 1 for any x; on the other hand p1 is a periodic point of period n1 1/Ex s with L (p1,f1)=−λ1 and so, for some neighborhood U of the orbit of p1 we have −˜ n1 ≤ ( λ1+)n1 − that Df s e where is such that λ2 + <0 : finally perform the 1/Ex perturbation in a tiny neighborhood of the orbit of p1 such that any point outside U is outside the support of the perturbation in the next m iterates. The above − ∈ n1 ≤ ( λ2+)n1 ∈ implies that f2 is Anosov, since if x U then Df s e and if x/U 2/Ex m m the Df s = Df s < 1. 2/Ex 1/Ex One last remark on how is chosen the αn = distC0 (fn,fn+1)(whereαn plays ≤ theroleforfn as α1 with f1): n>m αn αm. Therefore, inductively we have a sequence fn of Anosov diffeomorphism on 2 γn T such that distC1 (fn,fn+1) ≤ 2(e − 1) ,distC0 (fn,fn+1) ≤ αn,andfn has a periodic point pn whose stable Lyapunov exponent is −λn. The sequence fn is also uniformly dominated (i.e same cones and estimates) and the foliation tangent to s ∞ { } 1 EA is kept invariant. Since γn < we have that fn is a C -Cauchy sequence and thus converges to a C1 diffeomorphism g, having a dominated splitting on T2. The diffeomorphism g is not Anosov since g has a sequence pn of periodic points whose (stable) lyapunov exponent converges to zero. Every periodic point of g is hyperbolic: let q be periodic of period k, then for n ≥ k we have that q is disjoint of the support of perturbation of fn (since dC0 (g, fn) ≤ αn)andsog = fk along the orbit of q andsoitishyperbolic.Itisnotdifficulttoseethatg is expansive and so it is conjugated to Anosov [Le] (or see that the lift of g to R2 has infinity as expansivity constant). This concludes the proof of Theorem 6.5. Remark 6.1. The above example does not satisfy certain C1 generic conditions. For instance, it has no periodic attractor but by an arbitrarily small perturbation one can create them. Notice also that by multiplying this example with a strong contraction we have a diffeomorphism on a manifold having a homoclinic class with dominated splitting which is not hyperbolic and by perturbation we can not have a periodic point in the class of different index! (recall Problem 3). However, the homoclinic class can be perturbed to be hyperbolic Besides, it also gives an example of a homoclinic class s c u c H(p) with dominated splitting TH(p)M = E ⊕E ⊕E where E is not contracted but no periodic point in the class has index dimEs (see Problem 4). Taking into the account the results of this section, one may ask: does smoothness has a role to play in Problems 4 and 6?

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CMAT, Facultad de Ciencias, Universidad de la Republica´ del Uruguay

Igua 4225. Montevideo, Uruguay E-mail address: [email protected]

Contemporary Mathematics Volume 656, 2016 http://dx.doi.org/10.1090/conm/656/13075

Geometric regularity estimates for elliptic equations

Eduardo V. Teixeira

Abstract. In this article I present a geometric approach for establishing regularity estimates of solutions of diﬀusive equations. After discussing the heuristics of the method and few examples, I prove an improved regularity estimate along the set of critical points of solutions to elliptic equations in divergence form and a sharp growth estimate for dead-core problems ruled by the p- Laplace operator.

1. Overview Regularity theory for diffusive operators is among the finest treasures of the modern mathematical sciences. It appears in several different fields, such as, differential geometry, topology, numerical analysis, dynamical systems, mathematical physics, economics, etc. Within the general field of Partial Differential Equations, regularity theory places itself in the very core, by bridging the notion of weak solutions (often found by energy methods or probabilistic interpretations) to the classical concept of solutions. Probably the first contact undergraduate students have with regularity theory and its theoretical manifestations is in the complex calculus course. Along the first few lectures, the instructor defines the notion of holomorphic functions, which involves one complex derivative. Few lectures later, holomorphic functions are proven to be analytic, in particular of class C∞ in its domain of definition. What is even more surprising is the fact that one can bound universally all the derivatives of holomorphic functions, just by controlling their L2 norms. Yet at the beginning of that course, the students learn about the Cauchy-Riemann equation which provides a necessary and sufficient condition for a differentiable function f(x, y)tobe holomorphic. A consequence of such an observation is that, at least in simply con- nect domains, a function f(x, y)=u(x, y)+iv(x, y) is holomorphic if, and only if, both u and v are harmonic functions, that is: Δu =Δv = 0, where Δ denotes the Laplacian operator, Δu := ∂xxu + ∂yyu. Few courses down the road, the students finally meet the multi-dimensional version of the Laplacian operator in their introductory course on partial differential equations: if u is twice differentiable in an open set of Rn, then its Laplacian is

2010 Mathematics Subject Classiﬁcation. Primary 35B65.

c 2016 American Mathematical Society 185

186 EDUARDO V. TEIXEIRA defined as ··· (1.1) Δu := ∂x1x1 u + ∂x2x2 u + + ∂xnxn u, ··· where ∂xkxk ,fork =1, 2, n, denotes the pure second derivative in the kth- direction. Surprisingly enough, a similar regularity phenomenon, as in the complex calculus course, holds true. If u is harmonic, in the sense that u verifies Δu =0, then u is real analytic. Furthermore, one controls all derivatives of any harmonic 2 n function by the L norm. That is, if u is harmonic, say in B1 ⊂ R ,then α ∞ ≤ · 2 D u L (B1/2) Cn,|α| u L (B1), where Cn,|α| > 0 depends only on dimension n and the order of the derivatives |α|. The Laplacian operator (1.1) appears in several mathematical models coming from very different backgrounds. From the physical perspective, such an operator represents diffusion. For sake of illustration, let us briefly revisit the theory of elastic membranes with forcing terms (wind, weight, gravitation, etc. . . ). Physi- cal considerations and some mathematical simplifications drive us to the following minimization problem: (1.2) min |∇u|2 + f(X) · u dX u = g on ∂Ω , Ω where Ω is a bounded domain of Rn, g represents an eventual deformation (twist) of ∂Ωandf is the forcing term. Now let ϕ be your favorite smooth function that vanishes on ∂Ω and consider competing function vt(X):=u(X)+tϕ(X). Since u (the position of the membrane) is a minimum point of the functional (1.2), elementary Calculus implies that d |∇ |2 · vt + f(X) vt dX =0, dt Ω t=0 ∇ ·∇ · which readily reveals that Ω u ϕ + f(X) ϕdX = 0. Integrating by parts and using the fact that ϕ was taken arbitrarily, yields (1.3) −Δu = f(X), in Ω, at least if u has enough derivatives to justify the steps above. That is, the membrane satisfies a non-homogeneous Laplace equation. A key question in the theory of elastic membranes is to understand qualitatively the membrane deformation upon aforcingtermf. Intuitively, the elasticity properties of the membrane absorbs the forces and adjusts itself in a more organized fashion, as to keep minimizing the energy considered (tension, say). In a bit more precise mathematical terms, one is interested in understanding the regularity of u in terms of some lower order norms of f. Equation (1.3) is often called Poisson equation. Its regularity theory is part of a wider class of elliptic estimates called Schauder aprioriestimates, which assure that solutions to a linear, uniformly elliptic equation with C0,θ data, 0 <θ<1, i.e., functions u satisfying

(1.4) aij(X)Diju = f(X) where 0,θ (1.5) 0 <λ≤ aij(X) ≤ Λ,aij,f ∈ C ,

GEOMETRIC REGULARITY ESTIMATES FOR ELLIPTIC EQUATIONS 187 are locally of class C2,θ. Furthermore, there exists a constant C>0, depending only upon dimension, ellipticity constants (λ, Λ), and the θ–H¨older continuity of the data, aijC0,θ and fC0,θ , such that

2,θ ≤ · ∞ (1.6) u C (B1/2) C u L (B1). This is a fundamental estimate in the theory of PDEs and its vast range of applications. There are also Schauder aprioriestimates for equations in divergence form:

(1.7) div (aij(X)∇u)=f(X), under the same assumptions (1.5). Nonetheless, usually their regularity theory lacks one derivative: if u solves (1.7), under (1.5), then

1,θ ≤ · ∞ (1.8) u C (B1/2) C u L (B1). Before continuing, one should notice that estimate (1.6) is by no means a trivial fact. Even if one looks at the Poisson equation Δu = f(X), the Schauder estimate is saying that if the trace of the Hessian of function u is θ-Hölder continuous, for some 0 <θ<1, then all second order derivatives, even the ones that do not appear in the equation, like ∂x1x2 u is also θ-Hölder continuous for the same exponent θ. Intuition becomes even less precise when one looks at problems modeled in heterogeneous media. The mathematical formulation of such problems now involve variable coefficients, aij(X) as in (1.4) and in (1.7). In such context, it has been a common accepted aphorism that the continuity of the Hessian of a solution (in non-divergence form) or else the continuity of the gradient of a solution (in the divergence theory) could never be superior than the continuity of the medium. Notwithstanding, it has been recently established that such a phenomenon can occur, at least in particular meaningful points.

Theorem 1. Let aij(X) be a θ-H¨older continuous, uniform elliptic matrix, 0,θ i.e., 0 <λId ≤ aij(X) ≤ ΛId, with aij ∈ C (B1).Ifu veriﬁes

aij(X)Diju =0, in B1 − and Z is a zero Hessian point, i.e., D2u(Z)=0,thenu ∈ C2,1 at Z.If,onthe other hand, u solves a divergence for equation

div (aij(X)∇u)=0, in B1 − and Z is a critical point, i.e., ∇u(Z)=0,thenu ∈ C1,1 at Z. The proof of Theorem 1 is based on a rather powerful method based on “tangential regularity theories”. We will comment about these set of ideas in Section 2 and will survey about the result from Theorem 1 in Section 3.

Regularity theory for elliptic problems in discontinuous media, i.e., when aij(X) is merely elliptic, but with no further continuity assumptions, requires a whole new level of understanding and its mathematical treatment is profound. Such an issue appears for instance in the study of problems coming from Calculus of Variations: F(∇u) dX −→ min Ω where F : Rn → R is a convex function (19th Hilbert’s problem). It also appears in composite material sciences, where the diﬀusion process takes place within a

188 EDUARDO V. TEIXEIRA medium made from two or more constituent materials. This causes significant dif- ferences on physical and chemical properties along the medium. Each individual component of the material remains separate and distinct within the finished structure. From the macroscopic view point, we are led to the study of elliptic equations with coefficients given by: bij(X)= [aij(ω) · 1ω] ω∈B where B is a partition of Ω, each aij(ω) is uniformly elliptic, i.e., λId ≤ aij(ω) ≤ ΛId, and aij(ω) = aij(ω#), whenever ω = ω#. Another simple example of great interest in applied sciences are linear equations with homogenization process. It is given a smooth elliptic matrix aij(X)andone needs to look at limiting configurations as ε → 0 of solutions to elliptic equations (both in divergence and in non-divergence form) with coefficients given by " # X (1.9) b (X)=a . ij ij It is decisive then to obtain estimates that do not depend upon the homogenizing parameter >0. Hence, it requires estimates that do not depend on continuity assumptions of the coefficient matrix. Universal continuity estimates for solutions to divergence form equations is the contents to the nowadays called De Giorgi-Nash-Moser regularity theory. Twenty years past until Krylov and Safonov established the non-divergence counterpart of such a result. Through such eruditions, we now know that any solution to any elliptic equation has a universal modulus of continuity. In principle such results can be understood as aprioriestimates. However, it is possible to establish the results using the language of weak solutions, i.e., H1 distributional solutions in the divergence form equations and the theory of viscosity solutions for non-variational problems. Even though these original results are for linear equations, they are indeed non-linear devices. For instance, Krylov-Safonov Harnack inequality unlocks the theory of fully nonlinear elliptic equations. If Sym(n) denotes the space of n × n symmetric matrices, an operator F :Sym(n) → R is said to be elliptic if λP ≤F (M + P ) − F (M) ≤ ΛP , for all P ≥ 0. If u is a viscosity solution to a fully nonlinear elliptic equation F (D2u)=0, then, at least heuristically both u and uν satisfy a uniform elliptic equation, for any directional derivative uν . Hence, by the Krylov-Safonov Harnack inequality, both 0,β u and uν are of class C for some 0 <β<1. If in addition the operator F is assumed to be convex, then a Theorem due to Evans [8]andKrylov[10] separately implies that u is indeed C2,α. The question on whether solutions to fully nonlinear elliptic equations are of class C2 challenged the community for more than twenty years. It was answered in the negative by Nadirashvili and Vladut, [11, 12], who exhibit solutions to uniform elliptic equations whose Hessian blows-up. As much as Krylov-Safonov regularity theory yields the study of fully nonlinear elliptic equations, the corresponding De Giorig-Nash-Moser theorem was a starting

GEOMETRIC REGULARITY ESTIMATES FOR ELLIPTIC EQUATIONS 189 point for the development of degenerate quasilinear elliptic equations of the p- Laplace type, back in late 1960’s: p−2 Δpu := div |∇u| ∇u .

The p-Laplacian operator Δp appears naturally in the mathematical formula- tions of a number of physical problems. One can see it as a nonlinear version of the Laplacian (notice that when p = 2 be get back the original Laplace operator). The smoothing effects of the p-Laplace operator is less efficient than its linear mem- ber, the Laplacian Δ, but not because of nonlinear effects, but actually because the ellipticity of the operator degenerates as the gradient vanishes. It is known that alway from the set of critical points, C(u):={X ∇u(X)=0}, p-harmonic functions, i.e., solutions to

(1.10) −Δpu =0 are in fact quite smooth – real analytic. That is, the villain of the theory is precisely the a priori unknown set of critical points. The first major result in the area is due to Uraltseva, who proved in [23], for the degenerate case, p ≥ 2, that p-harmonic functions, are locally of class C1,α for some exponent 0 <α<1 that depends on 1,αp p and dimension. Nevertheless, Cloc is indeed optimal, since along its singular set C(u), p-harmonic functions are not, in general, of class C2, nor even C1,1.The optimal exponent for gradient Hölder continuity of p-harmonic functions are known only in dimension two, [9]. Nowadays, regularity theory for elliptic and parabolic problems is still a rather active line of research, that permeates several fields of mathematical sciences and applications. It is often regarded as a noble but difficult subject. It is present in a number of important recent breakthroughs as a decisive, key step. Through the next three Sections, we shall discuss a general geometric approach for regularity issues that emerge in several different contexts. In Section 2 we will present a notion of tangential regularity theories, and in Sections 3 and 4 we will exemplify the powerful applicability of such a notion by establishing improved regularity estimates in two different scenarios.

2. Tangential regularity theories In this Section we shall discuss about a general, unifying method for approaching regularity issues in models involving diffusion processes. Generally speaking, different models involve different intrinsic structures which have direct influence on the smoothing effects of the operators associated to the corresponding mathematical problem. On the very top podium in the regularity hierarchy among all diffusive differential operators is the rich theory of harmonic functions. Infinite order aprioriestimates, maximum principles, Hanack inequality, Li- ouville theorems, monotonicity and frequency formulas, unique continuation principles, energy estimates, etc, are some of the many mathematical tools available in the study of problems ruled by the Laplace operator. It is often that the model considered is not as generous as the Laplace equation, but still some regularity theory is available. Tangential regularity theories refers to the heuristic idea of imagining the universe of all second order diffusive operators. Each model is represented by a dot. There are infinitely many paths between all the models, among themselves. Usually

190 EDUARDO V. TEIXEIRA the paths represent some sort of compactness for solutions to problems around the models to be linked. Of course one can always construct straight paths between two given operators, L0 and L1,namelyLt := tL1 +(1− t)L0. Depending on the aimed property to be proven for a given model, a specific path ought to be considered. Let us discuss a simple example as to clarify the ideas. Suppose we are willing to develop a regularity theory for the Poisson equation: (2.1) Δu = f(X), discussed in Section 1. Instead of looking at (2.1) as a non-homogeneous Laplace equation, we should understand it as a family of models within the universe of all second order diffusive operators; for each class of forcing terms f(X), we have one different model. The goal is to establish a regularity estimate for solutions to (2.1) upon the control on a given norm of the forcing term, which determines the class of forcing terms considered. Such a control will be converted into compactness, and it will allow us to understand the Laplace equation, Δh = 0 as the tangential equation for the problem, in the following way: if the norm of f is very tiny, then it means that the model is close (in that imaginary universe) to the harmonic functions representative. The program then is to import the rich tangential regularity theory (TRT) available for harmonic functions back to the original model, properly corrected through the path used to access that TRT. For instance, suppose that p f ∈ L (B1), with n/2 0, we define: 1 u (X):= u(λX). λ λ2−n/p Easily one verifies that

n/p Δuλ = fλ(X)=:λ f(λX),

p ≤ $ and that fλ L (B1) f p 1. In other terms, we have veriﬁed that uλ is also a solution to a model that is close enough to the harmonic functions representative. ∈ 0,α − Carrying this idea on, one proves that u Cloc (B1)forα := 2 n/p. Indeed, such a geometric approach to regularity extends in a natural way to establish the following sharp regularity estimate, see for instance [17, 18], for more general results: Theorem . ⊂ Rn ∈ p 2 Let u satisfy Δu = f(X) in B1 .Supposef Lweak(B1). Then , / n • ≤ 2 n/2 If p = 2 ,then u BMO(B1/2) Cn u L (B1) + f . , Lweak(B1) /

• ≤ p If n/2

• If p = n,thenuLogLip(B ) ≤ Cn uL2(B ) + fLn (B ) . 1/2 , 1 weak 1 /

• ∞ ≤ p If nn/2, the membrane absorbs and distributes such an impact in a way that it remains bounded. At the singular point, the membrane will develop a cusp of order |X|2−n/p.

GEOMETRIC REGULARITY ESTIMATES FOR ELLIPTIC EQUATIONS 191

It is possible to establish corresponding sharp parabolic estimates, both in space and in time, see for instance [21]. One can also obtain Lq estimates of u provided f ∈ Lp for p0 ei<0 ei>0 ei<0 where {ei :1≤ i ≤ n} are the eigenvalues of M ∈ Sym(n). Easily one notice that if λ → 1, Λ then the model converges to the harmonic functions representative (i.e. the dot that represents the space of all harmonic functions). Thus, one can interpret the Laplace equation Δu =0asthegeometric tangential equation of the manifold formed by − λ → fully nonlinear elliptic operators F as e := 1 Λ 0. In turn, at every scale it is possible to ﬁnd a harmonic function close to a viscosity solution to (2.2), provided the ellipticity aperture is small enough. Iterating such an argument gives that the graph of a viscosity solution u can be approximated by a quadratic polynomial with an error of order ∼ O ρ2+α , for any given α. This proves the following Theorem (which can also be understood as a consequence of Cordes-Nirenberg estimates):

Theorem 3. Let u ∈ C(B1) be a bounded viscosity solution of (2.2). Given ∈ − λ α (0, 1) there exists 0 = 0(n, α) > 0 such that, if 1 Λ <0, then u is locally of class C2,α and

2,α ≤ · ∞ (2.3) u C (B1/2) C u L (B1), for a universal constant C = C(n, α) > 0. Yet within the theory of fully nonlinear equations, we could ﬁnd another path to access the tangential regularity theory of harmonic functions. Indeed, given a fully nonlinear elliptic operator F , we could look at the family of elliptic scalings 1 F (M):= F (μM),μ>0. μ μ This is a continuous family of operators preserving the ellipticity constants of the original equation. If F is diﬀerentiable at the origin (recall, by normalization we always assume F (0) = 0), then → → Fμ(M) ∂Mij F (0)Mij, as μ 0. → In other words, the linear operator M ∂Mij F (0)Mij is the tangential equation of Fμ as μ → 0. Now, if u solves an equation involving the original operator F , 1 then uμ := μ u is a solution to a related equation for Fμ. However, if in addition

192 EDUARDO V. TEIXEIRA it is known that the norm of u is at most μ, then it accounts into saying that uμ is a normalized solution to the μ-related equation, and hence we can access the universal regularity theory available for the (linear) tangential equation by compactness methods. This proves the following Theorem (see [15], [4], [7]).

0 2 Theorem 4. Let u ∈ C (B1) be a viscosity solution to F (D u)=f(X) in B1, 1 0,α where F is a C , (λ, Λ)-elliptic operator and f ∈ C (B1),forsome0 <α<1. 1 There exist a δ>0, depending only upon n, λ, Λ,α,fα,andtheC norm of F , such that if sup |u|≤δ B1 2,α then u ∈ C (B1/2) and

2,α ≤ · u C (B1/2) M δ, 1 where M depends only upon n, λ, Λ,α,fα,andtheC norm of F .

One can also try to “import” regularity from the behavior of the operator at the inﬁnity. That is, depending on how the operator behaves for very large matrices should give information of C1,α estimates for solutions of the homogeneous problem F (D2u) = 0. This brings to the notion of the ressession function: F (M) := lim μF (μ−1M). μ→0 This notion also appears naturally in the study of free boundary problems governed by fully nonlinear operators, see [3, 14]. The idea is that the family Fμ(M):= −1 μF (μ M) forms a path of uniform elliptic operators (each Fμ is elliptic with the same ellipticity constants as F ), jointing F and F . Should F have a better − regularity theory, then one should be able to import it back to F ,uptoC1,1 .The following Theorem has been proven in [16]. Theorem 5. Let F be a uniform elliptic operator. Assume any recession function F (M) := lim μF (μ−1M) μ→0 has C1,α0 estimates for solutions to the homogeneous equation F (D2v)=0.Then, any viscosity solution to F (D2u)=0, { }− 1,min 1,α0 ∈ 1,α { } is of class Cloc . That is, u Cloc for any α

1,α ≤ ∞ (2.4) u C (B1/2) C u L (B1), for a constant C>0 that depends only on n, α and F . An immediate Corollary of Theorem 5 is the following: Corollary 6. Let F : S(n) → R be a uniform elliptic operator and u avis- 2 cosity solution to F (D u)=0in B1. Assume any recession function F (M):= −1 1,α lim μF (μ M) is concave. Then u ∈ C (B1) for every α<1. μ→0 loc

GEOMETRIC REGULARITY ESTIMATES FOR ELLIPTIC EQUATIONS 193

As exemplified above, the method does not require the Laplace to be the tangential equation. For instance, one can study Poisson equations for more general operators, by interpreting the corresponding homogeneous problem as the tangential equation, [2, 5, 17, 18, 21]. Also, such an idea can be implemented in more adverse circumstances, such as in phase transmission problems, see for instance [1]. Another tempting path within the universe of all diffusive models one can consider is the p variationonthep-Laplacian operator. We conclude this Section explaining how the set of ideas implies the continuity of the underlying regularity theory for p−Laplacian operators with respect to p. This is done by the possibility of obtaining a universal compactness result. For instance, fix M0 % 2 and work within the range p ∈ [2,M0]. The following Lemma can be proven: Lemma 7 (Uniform in p compactness). Given δ>0,thereexists>0,depending only on n, M0 and δ, such that if q ∈ [2,M0], u is a q−harmonic function in B1,with|u|≤1,and|q − p| <, then we can find a p−harmonic function w in B1/2,with|w|≤1, such that (2.5) sup |w − u|≤δ. B 1 2 Let us comment on the proof of such a result: we suppose, for the sake of contradiction, that the thesis of the lemma does not hold true. This means that for a certain δ0 > 0, there exist sequences (qj)j ,(uj )j and (pj)j, with ⎧ ∈ ⎪ qj [2,M0]; ⎨ − div |∇u |qj 2∇u =0 in B ; (2.6) j j 1 ⎪ |u |≤1; ⎩⎪ j | − |≤ 1 pj qj j ; however for every pj −harmonic function w in B 1 , 2

(2.7) sup |uj − w| >δ0. B 1 2

By compactness, we have, up to subsequences, pj,qj → q∞ ∈ [2,M0], uj → u∞ locally in an appropriate functional space. By stability we can pass to the limit in the equation satisﬁed by the uj to conclude that u∞ is q∞−harmonic in B 2 .We 3 now solve, for each pj , the following boundary value problem pj −2 div |∇wj | ∇wj =0 in B 2 (2.8) 3 wj = u∞ on ∂B 2 3 and pass to the limit in j, concluding that also wj → u∞ uniformly in B 1 (by 2 uniqueness). Finally, choosing j large enough, we obtain

δ0 δ0 |uj − wj |≤|uj − u∞| + |wj − u∞|≤ + = δ0 in B 1 2 2 2 which is a contradiction to (2.7). One can use such a device to obtain improved sharp estimates for problems governed by p−Laplacian operators, near the Laplacian, i.e.,forp close to 2. After that, one can try to extend such estimates for a wider range, just by using now (2 + )-Laplacian as the tangential equation. Of course there is always the danger that at each iteration, the size of the step decreases in a summable fashion. As to

194 EDUARDO V. TEIXEIRA

have a conclusive argument to cover the whole range of exponents [2,M0] as finer analysis is required. In the next two Section we shall apply these general ideas explained here as to establish improved regularity estimates for two different models. The readers will be able to recognize the steps illustrated by the examples above. Each problem to be treated involves different tangential paths. In Section 3 we shall access the C2 aprioriestimates for harmonic functions with zero first order approximations; whereas in Section 4 we will make use of maximum principle tools available for the geometric tangential equation of the model as to establish a surprising gain of smoothness for problems ruled by the p-Laplacian along a particularly important region.

3. Improving Schaulder estimates In this Section we shall proof the divergence part of Theorem 1. In fact this is a consequence of a deeper Theorem established in [19]. The proof presented here though will be simpliﬁed and hence more appealing from the didactical point of view. 1 The initial set-up is the following, we have a function u ∈ H (B1) satisfying

(3.1) div (aij(X)∇u)=0, in the distributional sense. The coefficients are assumed to be uniform elliptic and θ-Hölder continuous, for some 0 <θ$ 1. From Schauder classical Theorem, u is locally of class C1,θ, hence we can talk about ∇u at a given interior point. We will focus our analysis at a critical point of u, i.e., a point Z ∈ B1, with ∇u(Z)=0. Let us open the floor by explaining the heuristics behind such a (surprising) result. If one is trying to show that an arbitrary given function u is of class C1,α at, say, the origin, the task is to find an affine function (X) that approximates u up to an error of order O(r1+α). If 0 happens to be a critical point for u, i.e., 0 ∈ C(u), then the 1st order of the approximation should be zero, and we are led to control the oscillation balance of u around a real constant. Now this is in accordance to the general (still heuristic) fact that even though elliptic equations in divergence form are 2nd order differential operators, in fact it reflects an oscillation balance around constants, rather than affine functions, as in the non-divergence theory. That explains why in many situations the regularity theory for divergence form operators has one derivative less than the non-divergence one. Continuing with the set-up, we notice that since aij is continuous, up to a zoom-in, we can assume that |aij(X) − aij(0)| is as small as we wish. Also, up to a change of variables, we can assume, with no loss, that aij(0) = δij (the identity matrix). Here it is the first key observation: 1 Lemma 8. Let u ∈ H (B1) be a weak solution to (3.1),normalizedasto 2 |u| dX ≤ 1 and aij 0,θ ≤ 1. Then, given a number δ>0,thereexistsa B1 C constant ε>0, depending only on δ, n, λ, Λ, such that if ∇u(0) = 0 and

(3.2) |aij(X) − δij|≤ε

then there exists a harmonic function h: B1/2 → R,forwhich0 isalsoacritical point, such that (3.3) |u − h|2dX ≤ δ2. B1/2

GEOMETRIC REGULARITY ESTIMATES FOR ELLIPTIC EQUATIONS 195

Proof. Let us assume, searching a contradiction, that the thesis of the Lemma 1 fails. That means that there exist a δ0 > 0, a sequence uk ∈ H (B1), with 2 (3.4) |uk(X)| dX ≤ 1, & ∇uk(0) = 0, B1 ≥ k for all k 1, and a sequence of elliptic (λ, Λ)-elliptic matrices aij with k ≤ k − → (3.5) aij C0,θ 1, & aij(X) δij 0, such that k ∇ (3.6) div aij(X) uk =0inB1; however | − |2 ≥ 2 ∀ ≥ (3.7) uk(X) h(X) dX δ0, k 1, B1/2 for any harmonic function h satisfying ∇h(0) = 0. From classical Schauder estimates (or even simpler energy estimates of Caccioopoli type) we deduce that there 1 exists a function u∞ ∈ H (B1/2) for which, up to a subsequence,

1 (3.8) uk u∞ in H (B1/2) 2 (3.9) uk → u∞ in L (B1/2)

(3.10) Duk(X) → Du∞(X)pointwiseforX ∈ B1/2, ∈ 1 Now, given a test function φ H0 (B1/2), in view of the approximation hypothesis (3.5), together with the limit granted in (3.8), we have ∇ ∇ k ∇ u∞, φ dX = aij(X) uk,Dφ dX +o(1) B1/2 B1/2 =o(1), as k →∞.Sinceφ was arbitrary, we conclude u∞ is a harmonic in B1/2.Also, by (3.10) together with assumption (3.4), we prove that 0 is a critical point of u∞. Finally, confronting such conclusion with (3.9) and (3.7), we reach a contradiction for k % 1. The Lemma is proven.

Notice that Lemma 8 has a universal character in the sense that it is true for any solution of any elliptic equation that are under the assumptions of the Lemma – the constant dependence is universal. This means that if we can rescale a given solution in a way to preserve the assumptions of Lemma 8, then the same conclusion holds for the rescaled function. Continuing inductively, we should then be able to import the regularity theory available for harmonic function back to our original function, properly readjusted by the scaling process. This is our strategy from now on. 1 2 Lemma 9. Let u ∈ H (B1) be a normalized (i.e., |u| dX ≤ 1) weak solution B1 to (3.1) with ∇u(0) = 0,whereaij is (λ, Λ)-elliptic matrix with aijC0,θ ≤ 1. 1 Then, given 0 <α<1, there exist constants 0 <ε0 < 1, 0 < < 2 , depending only upon n, λ, Λ and α, such that if

(3.11) |aij(X) − δij|≤ε0,

196 EDUARDO V. TEIXEIRA then, we can ﬁnd a universally bounded real constant τ ∈ R,i.e.,|τ|

Proof. For δ>0 to be chosen later, let h be the harmonic function in B1/2, with ∇h(0) = 0, that is δ-close to u in the L2-norm. The existence of such a function has been granted by Lemma 8. From C2 estimates on h, there exists a constant Cn depending only on dimension, such that |h(X) − h(0)|≤C|X|2. ∞ Since hL2 ≤ C,byL bounds, |h(0)|≤C. We now estimate, for >0tobeadjusteda posteriori, |u(X) − h(0)|2dX ≤ 2 |u(X) − h(X)|2dX + |h(X) − h(0)|2dX B B B 2 −n 4 ≤ 2δ +2Cn . Since 0 <α<1, it is possible to select small enough as to assure 1 2C 2(1+α) ≤ 4. n 2 Once selected ,asindicatedabove,weset

1 n +1+α δ := 2 , 2 which determines the smallness condition ε0, in the statement of this Lemma, through the approximation Lemma 8. The proof is concluded.

− We now conclude the proof of C1,1 regularity of solutions at critical points. Initially, if u solves div (aij(X)∇u)=0 inB1, the rescaled function u(γX) v(X):= , M satisfies γ−n/2 div (a (γX)∇v)=0 inB , & v 2 ≤ u 2 . ij 1 L (B1) M L (B1) 0,θ Now, if aij is C Hölder continuous at the origin and aij(0) = δij,wecanchoose γ $ 1sothat aij(γX)C0,θ ≤ 1& |aij(γX) − δij|≤ε0, where ε0 > 0 is the universal number from Lemma 9. Afterwards, we can choose M % 1sothatv is a normalized solution. That is, up to an scaling and a normalization, any solution to a divergence form elliptic equation with θ-Hölder continuous coefficients can be framed into the assumptions required by Lemma 9. So, we start off the proof out from (3.12). Next we rescale the function to the unit ball and normalize it, that is, we define u1 : B1 → R,by u( X) − τ u (X):= . 2 (1+α)

GEOMETRIC REGULARITY ESTIMATES FOR ELLIPTIC EQUATIONS 197

Easily one veriﬁes that u1 is under the very same assumptions of Lemma 9. Thus, there exists another universally bounded constantτ ˜ ∈ R, such that 2 2(1+α) |u2(X) − τ˜| dX ≤ , B which rescaling back to u gives 2 2(1+α) |u(X) − τ2| dX ≤ , B 2 where 1+α τ2 − τ =˜τ . In the sequel, we can deﬁne u( 2X) − τ u (X):= 2 , 3 2(1+α) and apply the argument again. Proceeding inductively, we obtain a sequence of real numbers τk, satisfying k(1+α) |τk+1 − τk|≤C , such that 2 k(1+α) |u(X) − τk| dX ≤ , B k In particular τk is a Cauchy sequence, hence it is convergent. From the inequality above, τk → u(0). Finally, given any 0 ˜ 0 that depends only upon universal parameters. The C1,α regularity of u at 0 follows. Notice that 0 <θ$ α<1 was taken arbitrary; hence − we have in fact proven that u is of class C1,1 at a critical point.

4. Improving regularity in p-dead core problems In this Section we turn our attention towards reaction-diﬀusion equations governed by p-Laplace operator (2 ≤ p<∞):

Δpu = f(u). Of particular interest are models coming from porous catalysis or enzymatic processes. A decisive aspect of the mathematical formulation of such problems is

198 EDUARDO V. TEIXEIRA the existence of dead-cores, i.e., regions where the density of the given substance vanishes identically. A prototype is the equation q (4.1) Δpu = λ0u , where λ0 > 0 is the Thiele modulus, which adjusts the ratio of reaction rate to diﬀusion–convection rate and 0 0}. We now state the key regularity improvement estimate we shall prove in this Section: Theorem 10 (Improved regularity). Let u be a nonnegative, bounded weak solution to (4.1) in B1 and let ξ0 ∈ ∂{u>0}∩B1/2 be a touching ground point. Then for any point X ∈ B1/2 ∩{u>0}, there holds

p − u(X) ≤ C|X − ξ0| p (q+1) , for a constant C depending only on dimension, p, u∞,andq.Inparticular p solutions are locally of class C p−(q+1) at any touching ground point. A warning should enhance the readers reaction towards Theorem 10: the touching ground surface lies in the set of critical points of solutions; which are precisely the points where the diffusion of the operator degenerates. It is then revealing to observe that, it follows in particular from Theorem 10 p − 2 that if dead-core exponent q>2 1, then solutions are C differentiable at any touching ground point. The proof will be based on a new flatness device, obtained by TRT methods, see [22]. Lemma 11 (Flatness estimate). Given η>0,thereexistsa = (η) > 0, depending only on η, dimension and p, such that if φ satisfies 0 ≤ φ ≤ 1, φ(0) = 0 and p q (4.2) Δpφ − δ φ =0, with 0 <δ≤ (η).Then,inB1/2, φ is bounded by η, i.e., (4.3) sup φ ≤ η. B1/2 Proof. Indeed, let us suppose, for the sake of contradiction, that the thesis of the flatness estimate Lemma fails to hold. That means that for some η0 > 0, there

GEOMETRIC REGULARITY ESTIMATES FOR ELLIPTIC EQUATIONS 199 exists a sequence of functions φk satisfying 0 ≤ φk ≤ 1, φk(0) = 0 and −p p Δpφk = k φk, in B1; however

(4.4) sup φk ≥ η0, B1/2 for all k ≥ 1. By the standard regularity theory for the p-Laplacian, up to a 1 subsequence, φk → φ∞ in the C topology. Clearly it veriﬁes

(4.5) φ∞ ≥ 0, (4.6) φ∞(0) = 0.

(4.7) Δpφ∞ =0, in B1. It then follows by the strong maximum principle that φ∞ ≡ 0insayB2/3. We reach a contradiction on (4.4) by choosing k % 1 large enough.

In the sequel, we shall apply Lemma 11 recursively in order to establish the improved regularity estimate along touching ground points. With no loss of generality we can assume ξ0 = 0. For positive constants 0 <κ, < 1, to be determined a posteriori, consider the normalized, scaled function

v(X)=κ · u( X), defined in B1. Easily one verifies that v solves , / (p−1)−q · p q (4.8) Δpv = κ v , in the distributional sense. Next we select 1 κ := , ∞ u L (B1) so that v is a normalized function in B1. It is time to choose and fix the value of . − p For that, we select η := 2 p−(q+1) and Lemma 11 sponsors the existence of > ˜ 0, such that for any function 0 ≤ φ ≤ 1, φ(0) = 0, satisfying p q Δpφ − δ φ =0, with δ ≤ ˜, we verify that − p (4.9) sup φ(X) ≤ 2 p−(q+1) . B1/2 With the value > ˜ 0 in hands, we decide that

−[p−(q+1)] := ˜ · κ p . − p With these choices, v is under the assumptions of Lemma 11, for η =2 p−(q+1) , i.e., − p sup v(X) ≤ 2 p−(q+1) . B1/2

In the sequel, let us deﬁne v2 : B1 → R,by

p X (4.10) v (X):=2p−(q+1) · v( ). 2 2

200 EDUARDO V. TEIXEIRA

One readily veriﬁes that

(4.11) 0 ≤ v2 ≤ 1;

(4.12) v2(0) = 0; p q (4.13) Δpv2 =˜ v2,

That is, v2 falls under the conditions requested by the ﬂatness estimate Lemma 11, which implies, − p − (4.14) sup v2(X) ≤ 2 p (q+1) . B1/2 Rescaling (4.14) back to v yields −2· p (4.15) sup v(X) ≤ 2 p−(q+1) . B1/4 Iterating inductively the above reasoning gives the following geometric decay: −k· p (4.16) sup v(X) ≤ 2 p−(q+1) . B 1 2k Finally, given any 0

Ackwnoledgement. This work has been partially supported by CNPq-Brazil, Capes-Brazil and Funcap.

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