Harmonic Forms, Minimal Surfaces and Norms on Cohomology of Hyperbolic 3-Manifolds

Xiaolong Hans Han

Abstract

We bound the L2-norm of an L2 harmonic 1-form in an orientable cusped hyperbolic 3-manifold M by its topological complexity, measured by the Thurston norm, up to a constant depending on M. It generalizes two inequalities of Brock and Dunﬁeld. We also study the sharpness of the inequalities in the closed and cusped cases, using the interaction of minimal surfaces and harmonic forms. We unify various results by deﬁning two functionals on orientable closed and cusped hyperbolic 3-manifolds, and formulate several questions and conjectures.

Contents

1 Introduction2 1.1 Motivation and previous results ...... 2 1.2 A glimpse at the non-compact case ...... 3 1.3 Main theorem ...... 4 1.4 Topology and the Thurston norm of harmonic forms ...... 5 1.5 Minimal surfaces and the least area norm ...... 5 1.6 Sharpness and the interaction between harmonic forms and minimal surfaces6 1.7 Acknowledgments ...... 6

2 L2 Harmonic Forms and Compactly Supported Cohomology7 2.1 Basic deﬁnitions of L2 harmonic forms ...... 7 2.2 Hodge theory on cusped hyperbolic 3-manifolds ...... 9 2.3 Topology of L2-harmonic 1-forms and the Thurston Norm ...... 11 2.4 How much does the hyperbolic metric come into play? ...... 14 arXiv:2011.14457v2 [math.GT] 23 Jun 2021 3 Minimal Surface and Least Area Norm 14 3.1 Truncation of M and the deﬁnition of Mτ ...... 16 3.2 The least area norm and the L1-norm ...... 17

4 L∞-norm, Main theorem and The Proof 18 4.1 Bounding L∞-norm by L2-norm ...... 18 4.2 Main theorem and the proofs ...... 21

1 5 Sharpness of the Inequalities and a Functional Point of View 22 5.1 Three conditions for the sharpness and their interactions ...... 23 5.2 Proof of non-sharpness using harmonic 1-forms and minimal foliations . . 24 5.3 A functional point of view ...... 26

References 27

1 Introduction

1.1 Motivation and previous results A cusped hyperbolic manifold is a complete non-compact hyperbolic manifold with finite volume. All manifolds are assumed oriented unless stated otherwise. By Mostow rigidity, the topology of a closed or cusped hyperbolic manifold M of dimension at least 3 determines its geometry. Effective geometrization attempts to seek qualitative and quantitative connections between the topological invariants and geometric invariants of the manifold. For example, given the fundamental group, what do we know about the injectivity radius, diameter, or 2-systole of the manifold (see [Whi02, BS11, BCW04, Bel13])? By a conjecture by Bergeron and Venkatesh, and similar conjectures independently pro- posed by Lück and Lêin [Lê18,Lüc16], one can extract the volume of a closed (arithmetic) hyperbolic 3-manifold, by the following:

Conjecture 1.1. ([BV13]) Let Mn be a sequence of congruence covers of a fixed arithmetic hyperbolic 3-manifold M. Denote the degree of the cover by [π1M : π1Mn]. Then we have log |H (M ) | vol(M) = 6π lim 1 n tor (1.1) n→∞ [π1M : π1Mn] where H1(Mn) is the abelianization of π1(Mn) and (·)tor is the torsion part of an abelian group. For recent progress and other related works on this conjecture, see [Le14, Lüc13, MM13, BD15, ABB+17]. In [BSV16], Bergeron, Sengünand Venkatesh have another striking conjecture that H2 of the Mn above can be generated by integral cycles in H2(Mn, R) with low topological complexity, as measured by the Thurston norm k · kT h (see Section 2 for detailed definitions). On the other hand, in a general closed Riemannian manifold 1 ∼ M, one can measure the complexity of a cycle in H (M; R) = H2(M; R) by the more geometric L2-norm: Z 1 kφkL2 = α ∧ ∗α, for φ ∈ H (M), M where α is the harmonic representative of φ from the classical Hodge theory. Bergeron, Sengünand Venkatesh proved a beautiful theorem regarding the two norms: Theorem 1.2. (BSV16, Theorem 4.1) If M varies through a sequence of finite coverings of a fixed closed orientable hyperbolic 3-manifold M0, then we have

C1 1 k · k ≤ k · k 2 ≤ C k · k on H (M; ) (1.2) vol(M) T h L 2 T h R

2 where C1 and C2 depend only on M0. The above theorem is another example of effective geometrization, where one bounds the geometric complexity of φ ∈ H1 by its topological complexity. In [BD17], Brock and Dunfield removed the assumption of Mn being covers of a fixed closed manifold M and obtained the following inequalities: Theorem 1.3. [BD17, Theorem 1.2] For all closed orientable hyperbolic 3-manifolds M one has π 10π 1 k · kT h ≤ k · kL2 ≤ k · kT h on H (M; R). (1.3) pvol(M) pinj(M)

Compared to [BSV16], Brock and Dunﬁeld introduced the tools of minimal surfaces and harmonic expansions of harmonic functions in H3 in the proof. In particular, the proof of the left-hand inequality uses the uniform bounds on the principal curvatures for a stable minimal surface in a closed hyperbolic 3-manifold and the Gauss-Bonnet theorem. It is interesting to note that the left-hand side has been approached by very diﬀerent techniques. Kronheimer and Mrowka and Lin use estimates for the solutions of the Seiberg-Witten equations in [KM97, Lin20]. Bray and Stern in [BS19] use the harmonic harmonic level set techniques, Bochner technique and the traced Gauss equations. The proof of the right-hand inequality combines several ideas. It uses explicit computations for the series expansions of harmonic functions and an inequality bounding L∞-norm of a 1-form in a ball by its L2-norm in the same ball, reminiscent of the mean value property of harmonic functions. Similar L∞ estimates were done by Cerbo and Stern in [CS19a, CS19b, CS20] for real and complex cusped manifolds with dimension at least 3 with implicit constants.

1.2 A glimpse at the non-compact case It is natural to wonder whether a version of Inequalities (1.3) still holds when M is cusped or infinite-volume. However, the L2-norm can depend on the geometry near infinity in a subtle way, probably not captured by the topology. The geometry of the cusped case is well-understood and much simpler than that of the infinite-volume case. Even if we only consider the cusped case, this question turns out to be nontrivial and interesting. Problems one naturally encounters include: 1. One immediate challenge is that the injectivity radius of M is zero and the right- hand side of Inequalities (1.3) is trivially true. We have to find the right geometric invariants in place of the injectivity radius. This issue will be resolved in a natural way after we have a deeper understanding of Hodge theory on non-compact hyperbolic 3-manifolds and minimal surface theory in this context. 1 ∼ 2. Originally, Inequalities (1.3) are for H (M) = H2(M) where M is closed. Brock and Dunfield use the stability of least area incompressible surfaces in closed hyperbolic 3-manifolds heavily, where one has uniform control on the principal curvatures on such surfaces and the area is bounded from above and below by some constant multiples of the Euler characteristic. In the non-compact case for H2(M), one can have closed surfaces homotopic to ones arbitrarily deep inside the cusps, and the

3 uniform control on curvature is no longer as clear. In addition, the Thurston norm k · kT h is no longer a genuine norm on H2(M) by the following example. Let T be a boundary-parallel torus. If M has more than one cusp, then [T ] 6= 0 in H2(M) but kT kT h = 0. In this case one can question how useful such an inequality is. Even if we restrict the consideration to closed surfaces with genus at least 2 in the thick part, it might have regions that can enter the cusp arbitrarily deep, resulting in unbounded area.

1.3 Main theorem It is perhaps surprising that the solution to all the issues mentioned above for a cusped M can be resolved by one simple condition: ψ ∈ H1(M): ψ can be represented by an L2 harmonic 1-form α. Denote by H1 the space of L2 harmonic 1-forms on M. In [Zuc83] (see also equation (1.4) and comments before that in [MP90]), Zucker showed that H1 is isomorphic to the 1 image of the natural inclusion of the compactly supported first cohomology Im(H0 (M) → H1(M)). We first provide a self-contained introduction to Hodge theory for non-compact hyperbolic manifolds in Section 2.2, based on [MP90], and then analyze the topology of the dual surfaces of the 1-forms in this space in Section 2.3, and show that the Thurston norm restricted to this space is an actual norm in Lemma 2.10. We also point out the beautiful coincidence that the dual surfaces are exactly those described recently in [HW17, Corollary 1.2]: Corollary 1.4. Let S be a closed orientable embedded surface in a cusped hyperbolic 3- manifold M 3 which is not a 2-sphere or a torus. If S is incompressible and non-separating, then S is isotopic to an embedded least area minimal surface. This corollary will provide us with enough regularity so that we obtain estimates similar to the ones for stable incompressible minimal surfaces in the closed hyperbolic case. In [HW17, Theorem 1.1], Huang and Wang show that for each cusped M, there is a canonical height τ to truncate cusps off M relative to the maximal volume cusp boundary to obtain a compact manifold Mτ with boundary so that any least area closed incompressible non- separating minimal surface stays in Mτ [HW17, Theorem 5.9, Corollary 5.7] (τ = τ3). One may naturally attempt to propose the following generalization of the Inequality (1.3) for a cusped 3-manifold M:

π c 1 p k · kT h ≤ k · kL2 ≤ p k · kT h on H (M) (1.4) vol(M) inj(Mτ ) where c is a explicitly computatble constant independent of M. Unfortunately, when we truncate the cusps, the boundary provides an essential term with a diﬀerent order of decay. However, using properties of subharmonic functions and modiﬁed Bessel functions, we manage to prove a version of the theorem independent of the truncation: Theorem 1.5. For all cusped orientable hyperbolic 3-manifolds M one has

r 2 π 10π d 1 k · kT h ≤ k · kL2 ≤ max{ , 4.86π 1 + }k · kT h on H , (1.5) pvol(M) psys(M) 2

4 where d is the largest intrinsic diameter of the boundary tori of the maximal cusp neighborhoods. Remark 1.6. In the proof we only use the fact that the smallest waist size of a cusp neighborhood is at least 1 [Ada02,Ada87]. In particular, the maximal cusp neighborhoods can be replaced by any cusp neighborhoods with waist sizes ≥ 1. Using Dehn ﬁllings, a natural corollary is that

Corollary 1.7. Let V0 be a ﬁxed positive constant. For all but ﬁnitely many cusped manifolds M with volume less than V0, we have 10π k · kL2 ≤ k · kT h (1.6) psys(M)

1.4 Topology and the Thurston norm of harmonic forms In Section2, we analyze the topology of the space of L2 harmonic 1-forms on an orientable 1 ∼ 1 cusped hyperbolic 3-manifold M, using the vector space isomorphism H = Im(H0 (M) → H1(M)) from [Zuc83,MP90]. We prove that this isomorphism is actually an isometry: Proposition 1.8. Let M be a cusped orientable hyperbolic 3-manifold. The space of L2 1 1 1 harmonic 1-forms (H (M), | · |L2 ) is isometric to (Im(H0 (M) → H (M)), k · kL2 ). In Section 2.3 we characterize the surfaces dual to H1 as the incompressible and non- 0 peripheral closed surfaces in M, which are the closest analog of the surfaces in H2(M ) 0 for M closed and hyperbolic. The Thurston norm on H2(M, ∂M) is a genuine norm. A natural corollary (Lemma 2.10) on the Thurston norm restricted to H1 is the following:

Lemma 1.9. Let M be a cusped non-compact orientable hyperbolic 3-manifold, k · kT h 1 ∼ on H = Im(H2(M) → H2(M, ∂M)) is a genuine norm. In general the Thurston norm is only a semi-norm. In Section 2.4, we also point out that the hyperbolic metric is important for Inequalities (1.3) and Theorem 1.5, as it might be impossible or useless to bound the L2-norm by the Thurston norm, when the Thurston norm fails to be a norm. For example, in the ﬂat 3-torus, such an inequality may behave like 0 ≤ 1 ≤ 0 (see Section 2.4). In elliptic manifolds, there are no incompressible surfaces by considering the fundamental groups, and there are no L2 harmonic 1-forms by Bochner type arguments. One interesting question is on which of the 8 geometries we have a useful inequality like Inequalities (1.3).

1.5 Minimal surfaces and the least area norm After analyzing the topology of the surfaces dual to H1, we show that the geometry of the surfaces dual to H1 is also very regular. In particular, their geometry works extraor- dinarily well with least area minimal surfaces. Existence of least area representatives of surfaces dual to H1 has been recently established in [HW17, CHMR17, CHMR19]. We show how to truncate the manifold M canonically to obtain a compact subdomain Mτ in Equation (3.3) (from [HW17]). All the least area surfaces dual to H1 are contained in Mτ (Proposition 3.2). Their areas are bounded from below and above by their genera

5 (Proposition 3.3), as in [Has95,CHMR17]. We then deﬁne the least area norm and show that it is controlled by the Thurston norm: Corollary 1.10. Let M be an orientable cusped hyperbolic 3-manifold. Then: 1 πk · kT h ≤ k · kLA ≤ 2πk · kT h on H (1.7)

We then define the L1-norm on H1 and show that it is equal to the least area norm in Lemma 3.6. This follow from the general principle that Poincaréduality is an isometry, with the natural Lp-norm and Lq-norm (Lemma 3.7), where 1 ≤ p, q ≤ ∞ are Hölder conjugates.

1.6 Sharpness and the interaction between harmonic forms and minimal surfaces It is a natural question whether Inequalities (1.3) and Inequalities (1.5) can be realized. We will prove that the left-hand of both are not realized. Theorem 1.11. Let M be an orientable, closed or cusped hyperbolic 3-manifold, then π kαkT h < kαkL2 pvol(M) for α ∈ H1(M) for closed M and α ∈ H1(M) for cusped M. For the closed case, the obstruction comes from the fact that a harmonic 1-form cannot have constant length. In particular, it follows from the fact that the second fundamental form of a minimal surface in a space form is the real part of a holomorphic quadratic differential, with 4g − 4 zeros, and cannot be everywhere nonzero. We provide two more proofs, one from [Zeg93] which proves the non-existence of smooth vector fields whose flow lines are all geodesic. The other is from non-existence of geometric foliations, from [WW20]. The non-compact case is surprisingly easier, due to the exponential decay of a harmonic 1-form near infinity (Theorem 2.6). We see a nice interaction between harmonic forms and minimal surface in Section5. We end by defining two functionals on Ψ := {Closed or cusped orientable hyperbolic 3-manifold} in Section 5.3.

πkαkT h Di(M) := inf 1 p α∈H vol(M)kαkL2 and

πkαkT h Ds(M) := sup p α∈H1 vol(M)kαkL2 Those functionals provide a unifying point of view for several results regarding sharpness and the examples constructed in [BD17]. It also enables us to ask questions not seen in the original inequality and may be an interesting invariant to study.

1.7 Acknowledgments The author would like to express a deep gratitude to his advisor Nathan Dunﬁeld for his extraordinary guidance and patience. The author thanks Zeno Huang, Richard Laugesen

6 and Franco Vargas Pallete for helpful discussions and emails, and especially for Pallete’s suggestion of looking at the zeros of the holomorphic quadratic diﬀerential in the proof of Proposition 5.4 and providing a reference by Mazet and Rosenberg as a proof of Corollary 5.7. This work is partially supported by NSF grant DMS-1811156, and by grant DMS- 1928930 while the author participated in a program hosted by the Mathematical Sciences Research Institute in Berkeley, California, during the Fall 2020 semester.

2 L2 Harmonic Forms and Compactly Supported Co- homology

2.1 Basic definitions of L2 harmonic forms Let M be a smooth Riemannian manifold of dimension n. We denote the space of smooth differential k-form on M by Ωk(M) and the subspace of forms with compact support by k Ω0(M). When the underlying manifold M is clear, we simply denote the two spaces by k k Ω and Ω0. The exterior derivative is denoted by d. Definition 2.1. The k-th de Rham’s cohomology group of M is defined by: {φ ∈ Ωk(M): dφ = 0} Hk (M) = dR dΩk−1(M)

k k By de Rham’s theorem, we will denote HdR(M) by H (M), the singular cohomology, where we always use R coefficients unless specified otherwise. Definition 2.2. The k-th de Rham’s cohomology group with compact support of M is defined by: k k {φ ∈ Ω0(M): dφ = 0} H0 (M) = k−1 dΩ0 (M) 2 Next we define L forms. Denote by dx1, dx2, ..., dxn a local orthonormal basis for the forms. A k-form φ locally can be expressed as P φ = φI dxi1 ∧ dxi2 ∧ ... ∧ xik I={i1Hilbert space with inner product Z Z hφ, βi = φ ∧ ∗β = (φ(x), β(x))gdvolg(x). M M k The formal adjoint of d on Ω0 is:

7 ∗ k+1 k d :Ω0 → Ω0 which is deﬁned by the following condition:

k+1 k ∀φ ∈ Ω0 and β ∈ Ω0, hd∗φ, βi = hφ, dβi Note that it can also be deﬁned on k-forms using the Hodge star operator by d∗φ = (−1)n(k−1)+1 ∗ d ∗ φ. The Hodge Laplace operator

∗ ∗ k k ∆ = dd + d d :Ω0 → Ω0 (2.1) is a second order symmetric elliptic operator. When the metric is complete, by [Gaf55], the L2 closure of the operator ∆:

∆ : L2Ωk → L2Ωk (2.2) is essentially self-adjoint. Its domain is {ω ∈ L2Ωk : dω, d∗ω, dd∗ω, d∗dω ∈ L2}. Definition 2.3. The L2 harmonic k-forms are defined by Hk(M) = {φ ∈ L2Ωk(M) : ∆φ = 0} where ∆ = dd∗ + d∗d. The above definition is equivalent to Hk(M) = {φ ∈ L2Ωk(M): dφ = d∗φ = 0} when the metric on M is complete by [Gaf55]. Using the Hodge-de Rham decomposition of L2Ωk(M) (also called Kodaira decomposition) [Car07,dR84]:

2 k k k−1 ∗ k+1 L Ω (M) = H (M) ⊕ dΩ0 (M) ⊕ d Ω0 (M). Deﬁnition 2.4. Relative and absolute boundary conditions for forms: Consider a manifold M with boundary ∂M, with ι : ∂M → M is the natural inclusion map. For a k-form α, the relative boundary condition is ι∗α = 0, while the absolute boundary condition is ινα = 0, where ιν is the interior multiplication with the inward pointing unit normal vector ﬁeld ν on ∂M.

The relative boundary condition resembles Dirichlet boundary condition f|∂M = 0 for functions, while the absolute boundary condition resembles Neumann boundary condition ∂f ∂ν |∂M = 0, where ν is the unit normal vector ﬁeld on ∂M pointing outward. When M is the interior of a compact manifold M with compact boundary ∂M: M = M − ∂M, the k kth cohomology with compact support H0 (M) is isomorphic to the relative cohomology group of M:

{φ ∈ Ωk(M): dφ = 0, ι∗φ = 0} Hk(M) = Hk(M, ∂M) := (2.3) 0 {dψ ∈ Ωk−1(M): dψ = 0, ι∗ψ = 0} where ι : ∂M → M is the inclusion map.

8 2.2 Hodge theory on cusped hyperbolic 3-manifolds Throughout the paper M is an orientable closed or cusped hyperbolic 3-manifold, unless noted otherwise. When M is cusped, it is the interior of an orientable irreducible compact 3-manifold M with toroidal boundary. The boundary of M refers to ∂M. This is coincident with the compactification of cusped hyperbolic manifold as in [MP90]. This subsection covers Hodge theory on cusped hyperbolic 3-manifolds, in particular for the space of L2 harmonic 1-forms H1. Most ideas are adapted from [MP90], which builds a Hodge theory for geometrically finite hyperbolic n-manifolds with possibly infinite volume, generalizing the results on cusped manifolds from [Zuc83], and obtains asymptotic estimates for L2 harmonic k-forms near infinity. The particular case that we are interested in is the following from [MP90,Zuc83]: 1 ∼ 1 Lemma 2.5. Let M be an orientable cusped hyperbolic 3-manifold. Then H = Im(H0 (M) → 1 1 H (M)) as vector spaces, where H0 (M) is the first cohomology with compact support. From [MP90], the isomorphism can be interpreted in the following way. Take α ∈ H1, 1 by doing a perturbation supported near infinity, we obtain φ ∈ H0 such that α is cohomologous to φ in H1. The specific construction is in Proposition 2.7 when we promote 1 the isomorphism above to an isometry. From basic algebraic topology, H0 (M) is isomorphic to H2(M) by [Hat02, Theorem 3.35]. There is a subtle but important difference 1 1 1 between Im(H0 (M) → H (M)) and H0 (M), which turns out to be crucial for the main theorem. Consider a boundary-parallel torus T in a cusp. It is clear that T ∈ H2(M), and a 1-form α dual to T can be chosen to be compactly supported around T . The 1-form α has Thurston norm zero, and it is not obvious that its harmonic norm is zero. Moreover, as it exits to infinity, its area is approaching zero. Fortunately, [T ] = 0 in H2(M, ∂M) and hence [T ] = 0 in Im(H2(M) → H2(M, ∂M)), equivalent to [α] = 0 in 1 1 2 Im(H0 (M) → H (M)). We can also see this from a computation of L -norm. Place T as a quotient of the horosphere z = c in the upper half-space model. A 1-form dual to T is α = f(z)/zdz where f(z) is a bump function supported in [c − 1, c + 1] with R f/zdz = 1. Z Z c+1 2 2 dx ∧ dy ∧ dz c + 1 2 O |α|L = α ∧ ∗α = f 3 = log → 0 as c → ∞ M c−1 z c − 1 The third equality above is a simple estimate from R f/zdz = 1. If [α] 6= 0 in H1(M), one of the components of the closed surface dual to α must have genus at least 2, as there are no essential spheres or essential tori. For non-compact properly embedded incompressible surfaces, we will show they are dual to 1-forms with infinite L2 norm. We also need an asymptotic control on the L2 harmonic 1-forms that is developed in [MP90]. We reformulate their Theorem 4.12 into the one we need: Theorem 2.6. If α ∈ H1, then on a neighborhood of a rank-2 cusp α = φ + βdz where |φ| = O(e−λz) and β = O(e−λz) for some λ > 0, where φ is a 1-form and β is a function, both depending parametrically on z, the height coordinate in the upper half space model of H3. Essentially, the idea of the exponential decay is the following. One writes down the Laplacian equation for the 1-forms in the cusps, using separation of variables. Then

9 this is equivalent to solving some Fourier-Bessel equations, whose solutions either grow exponentially or decay exponentially. See also [Dod82]. Since α ∈ L2, it must decay exponentially. For people familiar with number theory, this is consistent with, for exam- 1 ∼ 1 1 ple, [EGM98, Theorem 3.1]. The isomorphism H = Im(H0 (M) → H (M)) is actually 1 1 2 an isometry. Equip Im(H0 (M) → H (M)) with a natural L -norm by the following deﬁnition: 1 1 kφkL2 := inf{|ψ|L2 | ψ ∈ H0 , cohomologous to φ in H }. (2.4)

Proposition 2.7. Let M be a cusped orientable hyperbolic 3-manifold. The space of L2 1 1 1 harmonic 1-forms (H (M), | · |L2 ) is isometric to (Im(H0 (M) → H (M)), k · kL2 ).

Proof. The proof follows from the proof of [MP90, Theorem 3.13]. Fix a cusp neighborhood N = [t, ∞) × T2, with coordinate (y, x). The first step is to compactify the cusp at infinity and introduce s = 1/y as new coordinate. Now N = [0, ) × T2 for some small > 0. The compactification is actually a natural and powerful way to establish theorems for some of the non-compact manifolds. Now M is treated as a compact manifold with boundary, with s = 0 corresponding to ∂N. Near ∂N or s = 0 a harmonic 1-form α can be written as α = γ(s) + h(s)ds (2.5) where h ∈ C∞(M). Here γ, h should be treated as forms and functions depending parametrically on s. By [MP90, (3.3)], we have the following retraction operator,

Z s Rα(s) = h(t)dt. (2.6) 0 If α is a 1-form, Rα is a smooth function. One can verify that near s = 0, one has

α = dRα + Rdα + γ(0) (2.7)

The second term Rdα is defined similarly to Rα for a 1-form α, where we decompose it into components without and with ds, and then integrate. Since α is harmonic dα = 0, and α = dRα + γ(0). Take a cutoff function f(s) equal to 1 near s = 0, supported in N. Now f(s)Rα is globally defined. With i : ∂N → N the inclusion, i∗(f(s)Rα) = 0 and α − d(fRα) vanishes near ∂N. We apply the above construction in each of the cusp components. The map

1 1 1 F : H → Im(H0 (M) → H (M)),F (α) = α − d(hRα) (2.8) is an isomorphism as proven in [MP90, Theorem 3.13]. We will use functions fi with increasingly small support to construct a sequence of smooth forms with compact support αi such that |α − αi|L2(M) → 0 (2.9) By computation, near s = 0

Z 2 1 2 3 |α|L2 = |γ(s)| + |h(s)| s ds 0 s

10 2 2 If α ∈ L forces γ(s) = γ(0) = 0 near s = 0. Define Ni := [0, i ) × T , fi is a cutoff i function equal to 1 near s = 0 with supp(fi) ⊂ Ni and |dfi| ≤ . Define

αi = α − d(fiRα) (2.10) which implies

Z /i Z s 2 2 3 2 2 2 |α − αi|L2 = |d(fiRα)|L2 = s |∂sfi(s)| h(t)dt + |fi(s)| |h(s)| ds 0 0 Z /i 2 Z s 2 3 i 2 ≤ s · 2 h(t)dt + |h(s)| ds 0 0 3 i2 ≤ C → 0 i3 2 as i → ∞, where we use h ∼ O(e−λ/s) by Theorem 2.6.

2.3 Topology of L2-harmonic 1-forms and the Thurston Norm

1 ∼ 1 ∼ By Equation (2.3), H0 (M) = H (M, ∂M) = H2(M). It is a classical fact that every nontrivial [S] ∈ H2(M) for both closed and cusped M can be represented by closed smoothly embedded incompressible surface. This follows from Poincar´eduality and regular value theorem [Thu86]. We will henceforward assume every class in H2 is represented by such a closed smoothly embedded incompressible surface. Deﬁnition 2.8. A surface S is peripheral if it is homologous to a union of boundary components. A surface is non-peripheral if it is not peripheral.

When M is cusped, a nontrivial class [S] ∈ H2(M) can be represented by a peripheral surface, where each component of M − S contains a boundary component of M, or non- peripheral ones. The main reference for the Thurston norm is Thurston’s original paper [Thu86]. The Euler characteristic is negative for surfaces with genus ≥ 2, and it is more convenient to use its absolute value to measure the topological complexity. We deﬁne for a connected surface χ−(S) = max{−χ(S), 0}, where χ is the Euler characteristic. Extend 0 0 this to disconnected surfaces via χ−(S tS ) = χ−(S)+χ−(S ). For a compact irreducible 1 ∼ 3-manifold M, the Thurston norm of α ∈ H (M; Z) = H2(M, ∂M; Z) is deﬁned by

kαkT h = min{χ−(S)|S is a properly embedded surface dual to α} The Thurston norm extends to H1(M; Q) by making it linear on rays through the ori- gin, and then extends by continuity to H1(M; R). When M is closed and hyperbolic, it is non-degenerate and a genuine norm. In general it is only a semi-norm, for ex- 1 ∼ ample, for H0 (M) = H2(M) when M is a cusped hyperbolic 3-manifold, due to the presence of incompressible boundary-parallel tori. However, the story is different for 1 ∼ H = Im(H2(M) → H2(M, ∂M)). We first give a classification of nontrivial classes in H2(M; Z): 1. Such a class [S] can be peripheral. In this case, there exists a proper map u : (M, ∂M) → (I, ∂I), where I is a closed interval. If we denote by [dx] a generator

11 1 ∗ for H0 (I), then the 1-form α dual to S satisﬁes [α] = [u (dx)]. We also have [S] is homologous in H2 to some boundary-parallel tori, and hence represents the 1 trivial class in H2(M, ∂M). Thus a nontrivial α ∈ H cannot be dual to a closed incompressible peripheral surface.

(a) Peripheral (b) Non-peripheral

Figure 1: Classiﬁcation of H2(M)

∼ 2. The class [S] can also be non-peripheral, and this is similar to the case of H2(M, ∂M) = [M : S1], where it is the fiber of a regular value of a smooth map u : M → S1. If we take u to be harmonic, or the energy-minimizing representative in its homotopy class, the harmonic 1-form α = u∗(dθ) satisfying the relative boundary condition (Definition 2.4) is dual to S, where [dθ] is a generator for H1(S1). A nonzero L2 harmonic 1-form is in particular represented by a non-peripheral surface. Conversely, for a non-compact incompressible surface S with finite topology whose ends are properly embedded, like a thrice punctured sphere embedded inside cusped hyperbolic 3-manifolds [Ada85], then the L2-norm of its dual 1-form α must be infinite. This is well- known to the experts, but we provide a short proof for completeness.

Lemma 2.9. If [S] ∈ H2(M, ∂M) − H2(M), then a harmonic 1-form [α] dual to S does not have ﬁnite L2-norm.

2 1 Proof. If the 1-form α dual to S has ﬁnite L -norm, by Proposition 2.7 and Im(H0 (M) → 1 ∼ H (M)) = Im(H2(M) → H2(M, ∂M)), S must be homologous to the image of a class 0 [S ] ∈ H2(M), a contradiction.

We can compute the lower bound of the growth rate, using compactification and change of coordinate s = 1/z and α = γ(s) + h(s)ds, where s = 0 corresponds to infinity in R 2 1 a cusp component, as in [MP90, (3.7)]. We have kαkL2 ≥ s kγ(0)k t dt = − log s. An elementary computation of the growth rate upper bound is the following. From [CHR18], we know the ends of such a surface are asymptotic to totally geodesic infinite annuli inside the cusps. It suffices to do the computation on one of its ends asymptotic to a truncated cusp Ci. For simplicity, first assume there is exactly one end of the surface inside Ci and 3 the base of Ci is the unit square. Parameterize Ci by {{(x, y, z) ∈ H :(x, y) ∈ [0, 1] × [0, 1], z ≥ c}}. Since the end is an incompressible totally geodesic annulus inside the cusp,

12 we can suppose after change of coordinate this end is parameterized by {(1/2, y, z) ∈ Ci}. 2 2 In this case, at each horizontal torus Tc, kαk integrates to approximately c along a Euclidean geodesic intersecting the end exactly once and parameterized as {(x, 1/2, c); x ∈ [0, 1]}. Integrating in the z-direction inside the cusp, we have Z ∞ Z 1 Z 1 2 1 kαk 2 ∼ z dxdydz = ∞ L (Ci) 3 c 0 0 z In particular, it approaches ∞ like log z, consistent with the [BD17, Theorem 1.4]. A criterion for the Thurston norm to be a genuine norm is the following [Thu86]: For H2(M; Z) (or H2(M, ∂M; Z)), if every embedded surface representing a non-zero element has negative Euler characteristic, then k · kT h is a norm. All closed orientable surfaces have even Euler characteristic. When M is cusped, all incompressible tori are boundary- parallel and there is no incompressible surface with positive Euler characteristic. Thus it can be seen that

Lemma 2.10. When M is a cusped non-compact orientable hyperbolic 3-manifold, k·kT h 1 ∼ on H = Im(H2(M) → H2(M, ∂M)) is a genuine norm. Later we will use the L1-norm to mediate between the least area norm and the L2-norm. For the purpose of this paper, the norms are used for computations only and we will not use the full power of the Lp theory of forms, as in [Sco95]. The L1 and L∞-norms of an L2 harmonic 1-form α with are defined by Z |α|L1 := |α(x)|dvol and |α|L∞ = max |αp|. (2.11) M p∈M Since an L2 harmonic 1-form α decays exponentially near infinity, its L∞ and L1-norm are well-defined. The L∞ and L1-norm of a 1-form with compact support is defined similarly, and clearly finite. We also define similar functions on the classes of the cohomology space 1 1 Im(H0 (M) → H (M)):

1 1 1 kψkL1 = inf{|φ|L1 |φ ∈ H0 (M) represents ψ ∈ Im(H0 (M) → H (M))}. (2.12) When M is closed, the deﬁnition of L1-norm is just:

1 1 kψkL1 = inf{|φ|L1 |φ ∈ H (M) represents ψ ∈ H (M)}. (2.13)

The L1-norm behaves very differently compared to the L2-norm on the cohomology space, as it tends to be not realized by a smooth forms. A sequence of forms φi with compact support converging to φ realizing the L1-norm on the cohomology space behaves like convergence of a sequence of bumps functions to a Dirac functional. Later we will use Cauchy-Schwarz inequality to bound the L1-norm of α ∈ H1 by its L2-norm. To bound the L2-norm of α, we need to convert it into some integral of α on a surface S dual to α. When the manifold M is closed, it is clear by Poincaréduality that integrating against a closed integral 1-form is equal to integrating against its dual surface. Specifically, fix a surface S dual to φ ∈ H1(M; Z) and let α be the harmonic representative of φ. Then we R R have M β∧α = S β for every closed 2-form β. In the closed manifolds case, the harmonic form α is closed and coclosed. In the non-compact case, if the metric is complete, α ∈ H1 1 1 implies α is closed and coclosed. Let α ∈ H correspond to φ ∈ H0 (M, Z), dual to

13 [S] ∈ Im(H2(M) → H2(M, ∂M)). Fix a closed surface S dual to φ. By Poincar´eduality for forms with compact support [BT82, I.5], we again have for every closed 2-form β on R R M that M β ∧ φ = S β. We provide a short argument for the following lemma: Lemma 2.11. Let M be an orientable cusped hyperbolic 3-manifold. Let α ∈ H1, with a surface S dual to α. Then the following equation is true: Z 2 1 kαkL2 = ∗α for α ∈ H . (2.14) S

Proof. By continuity of the norms, it suﬃces to prove the equality for α ∈ H1 cor- 1 R R responding to φ ∈ H0 (M, Z). We have M ∗α ∧ φ = S ∗α and it suﬃces to show R ∗α ∧ (α − φ) = 0. Since [φ − α] = 0 in H1, we have φ − α = df for f ∈ C∞(M). Now M R if we can show M f ∗ α < ∞, then Z Z ∗α ∧ (α − φ) = d(∗α ∧ f) = 0, M M

by [Yau76, Lemma]. Applying the Cauchy-Schwarz inequality and the fact that α ∈ L2, we only need f ∈ L2. As φ − α = df and φ has compact support, we have df = α outside a large ball BR. Thus Z ∞ f 2 kfk 2 ∼ kfk 2 ∼ dz (2.15) L (M) L (M−BR) 3 c z

If kfkL2(M) = ∞, we have f grows at least like O(z), which contradicts the fact that df = α is decaying exponentially by Theorem 2.6.

2.4 How much does the hyperbolic metric come into play? After deﬁning all the norms, we address the short question of how much is a hyperbolic metric necessary for Inequality 1.3. Recall it is an attempt to study eﬀective geometrization, that is, for M cusped and hyperbolic, how is the geometry qualitatively and quantitatively determined by the topology. In contrast, consider a 3-torus M = T3 as in Figure2, from the gluing of three pairs of opposite faces of the unit cuboid in R3: {(x, y, z) ∈ R3|0 ≤ x, y, z ≤ 1}, where the vertical face {x = 0} is glued to {x = 1} and the vertical face {y = 0} to {y = 1}, and {z = 0} to {z = 1}. Consider the horizontal 1 torus S = {z = 2 }. It is clear that [S] ∈ H2(M) and kSkT h = 0. Since the metric on M is the Euclidean metric, it is clear that a harmonic 1-form dual to S is dz. Since |dz| = 1 and M has unit volume, we have kdzkL2 = 1 and the Inequalities 1.3 fail. One can construct many similar examples, where k·kT h fails to be a genuine norm and instead is only a semi-norm, and the work lies in showing the Thurston norms vanish for some of the nontrivial classes.

3 Minimal Surface and Least Area Norm

In this section, we discuss minimal surfaces and their properties in orientable closed and cusped hyperbolic 3-manifolds. A closed surface is minimal if its mean curvature is 0.

14 Figure 2: The essential torus S has Thurston norm 0 but L2 norm 1

Denote ν the unit normal vector field on a surface S with trivial normal bundle. Identify a normal vector field X = fν, where f is some smooth function. The stability operator L is defined as 2 Lf = ∆Sf + |σ| f + RicM (ν, ν)f. (3.1) where RicM is the Ricci tensor of the ambient manifold M, σ is the second fundamental form on the surface S, and ∆S is the Laplacian operator restricted to the surface. The definition of stability is from [CM11]. A minimal surface S is stable if for all compactly supported variations F with boundary fixed, d2 Z 2 Area(F (S, t)) = − hFt, LFtidA ≥ 0 dtt=0 S It is equivalent to the stability operator being negative semidefinite. A least area surface in a homotopy class is necessarily stable, but not vice versa. Let M be an orientable closed hyperbolic 3-manifold. A least area surface F ⊂ M with genus g ≥ 2 in its homotopy class satisfies the following inequality [Has95, Lemma 6], [MR17, Section 8.1] and [CHMR17, Remark 3]:

2π(g − 1) ≤ area(F ) ≤ 4π(g − 1) (3.2)

The left-hand side comes from the stability of F in a hyperbolic 3-manifold, while the right-hand side merely comes from the Gauss equation for a minimal surface. The regularity and properties enjoyed by a stable minimal (or least area) surface are in tune with Mostow rigidity for hyperbolic manifolds, in that its area can be determined by its topology. The existence of least area surface representatives in the homotopy class of [S] ∈ H2(M) for M compact is well-understood and can be approached by diﬀerent methods, from the geometric measure theory by Federer [Fed69] as early as 1969, the relatively geometric arguments of Freedman, Hass and Scott [FHS83] and the analytic arguments in [SU82]. The existence of least area incompressible closed surfaces in the cusped setting requires techniques not from usual minimizing techniques from the compact cases in [SY79, SU82], and was recently settled by Huang and Wang [HW17] and by Collin, Hauswirth, Mazet and Rosenberg [CHMR17], [CHMR19]. While M is topo- logically compact with boundary, it is geometrically non-compact and the surfaces can enter the cusps arbitrarily deep. Cusps share some similarity with short tubes as they can be foliated by Euclidean tori. This foliation is useful in some area estimates for cut and paste arguments. One obstruction for a least area closed surface to enter arbitrarily

15 deep inside the cusp is provided by the following argument. If the surface is sufficiently deep, the part inside the cusp will contribute to a large area. One can cut off the cusp at a torus at some height, obtain a surface with some boundaries components and glued the disks coming from the cut-off torus. This can reduce the area. Similar ideas are also used in [HW19] to investigate minimal foliation questions on hyperbolic 3-manifolds fibering over S1. Now the question is whether such an argument can provide us with uniform control on the closed incompressible surfaces in cusped M, independent of the surfaces. This is the content of: Theorem 3.1. [HW17, Corollary 1.2] Let S be a closed orientable embedded surface in a cusped hyperbolic 3-manifold M which is not a torus. If S is incompressible and non-separating, then S is isotopic to an embedded least area minimal surface.

3.1 Truncation of M and the deﬁnition of Mτ

Recall that a surface S ⊂ M dual to α is called taut if the surface S realizes kαkT h, is incompressible, and no union of components of S is separating. For the proof of the main theorem we are interested in estimating norms of α ∈ H1 and thus we will always 1 ∼ assume the dual surface S is taut. For H (M) = Im(H2(M) → H2(M, ∂M)), we will show there is a compact manifold Mτ with ﬂat toroidal boundary, containing all the least 1 area surfaces dual to H (M). We now describe the construction of Mτ as in [HW17]. Suppose M has k cusps. A cusp neighborhood is maximal if there are no larger cusp neighborhoods containing it, which occurs exactly when it is tangent to itself at one or more points. In this paper, the cusp neighborhoods are always cut out of by geometric horotori. Denote the maximal cusp neighborhoods by Ci = Ti × [0, ∞), i = 1, . . . , k (here the coordinate on [0, ∞) is the intrinsic geometric distance from Ti × {0}). Let τ0 > 0 be the smallest number such that each cusped region Ti × [τ0, ∞), i = 1, . . . , k is disjoint from any other maximal cusp region of M. For a constant τ ≥ τ0, let Mτ be the compact subdomain of M deﬁned by:

k Mτ = M − ∪i=1(Ti × (τ, ∞)) (3.3)

By construction, Mτ is a compact submanifold of M with concave boundary components with respect to inward pointing normal vectors. We lift M to H3 with the upper half space model to extract quantitative data of the boundary tori, essentially the translational distance corresponding to the parabolic isometry. For each i with 1 ≤ i ≤ k, we lift M in a way so that the horoball Hi corresponding to Ci is centered at ∞, and ∂Hi = {(x, y, z) ∈ 3 H |z = 1}. Suppose Γi is the rank-2 parabolic subgroup corresponding to Ci, generated by p 7→ p + ξi and p 7→ p + ηi, where ξi and ηi are non-trivial R-linearly independent complex numbers. Now deﬁne

τ0 L0 = max{e , |ξ1| + |η1|,..., |ξk| + |ηk|} > 0 and τ = log(3L0) (3.4)

Both constants are independent of the least area surfaces. Then Mτ contains all the least area surfaces described in Theorem 3.1 and τ is independent of the surfaces ([HW17, Remark 2.3]). We have the following: Proposition 3.2. Let α ∈ H1 be nontrivial. There is a least area representative S ∈ Im(H2(M) → H2(M, ∂M)) dual to α such that S ⊂ Mτ , where Mτ is a compact 3-

16 manifold with toroidal boundary truncated from M and τ is a constant independent of α. We can now control the geometry of the minimal surface by Proposition 3.2. Proposition 3.3. Let S be a closed, orientable, incompressible and non-peripheral embedded surface of genus ≥ 2 in an orientable cusped hyperbolic 3-manifold. The least area representative F in the homotopy class of S satisﬁes

2π(g − 1) ≤ area(F ) ≤ 4π(g − 1) (3.5)

The proof is in [CHMR17, Remark 3].

3.2 The least area norm and the L1-norm We are now ready to define the least area norm. We first define it with integral coefficient, 1 1 and then extend it by continuity. Denote {α ∈ H : α corresponding to φ ∈ H0 (M; Z)} by H1(M; Z). 1 Definition 3.4. For α ∈ H (M; Z), let Fα be the collection of smooth maps f : S → M where S is a closed oriented surface with f∗([S]) dual to α. The least area norm of α is

kαkLA = inf{Area(f(S))|f ∈ Fα} With this deﬁnition, a corollary of Proposition 3.3 is Corollary 3.5. Let M be an orientable cusped hyperbolic 3-manifold. Then:

1 πk · kT h ≤ k · kLA ≤ 2πk · kT h on H (M; Z) (3.6)

To bound k · kLA by k · kL2 , we need the following lemma. We remind the readers that the 1 ∼ 1 1 L -norm is merely a function on Im(H2(M) → H2(M, ∂M)) = Im(H (M) → H (M)), R 0 and in general not equal to M |α| for α harmonic. 1 1 1 Lemma 3.6. For φ ∈ H0 (M; Z), [φ] 6= 0 in Im(H0 (M; Z) → H (M; Z)), the least area norm and the L1-norm satisfy

kφkLA = kφkL1

Proof. For the full proof of the case when M is closed, see [BD17, Lemma 3.1]. We 1 mention some necessary changes for the non-compact case. For φ ∈ H0 (M; Z), the dual surface S is also smooth embedded, and can be chosen to be contained in Mτ . We can 1 construct a dual 1-form compactly supported near S, whose L norm very close to kφkLA gives an upper bound on kφkL1 . For φ dual to boundary-parallel tori exiting to ∞, it is clear that kφkLA = 0 = kφkL1 . To prove kφkLA ≤ kφkL1 it is slightly more complicated. However this is another case where our analysis of the topology of H1 simpliﬁes the 1 1 argument. The dual surface S is non-separating. If φ ∈ Im(H0 (M) → H (M)) is an integral class, by integrating φ we get a smooth map f : M → S1 so that φ = f ∗(dt), where S1 is parameterized by t ∈ [0, 1]. The remaining argument is identical.

17 1 With the above Lemma, the least area norm extends continuously from Im(H0 (M; Z) → 1 1 1 ∼ 1 1 H (M; Z)) to a seminorm on all of Im(H0 (M) → H (M)) = H . Let α ∈ H , and 1 φi ∈ H0 be a sequence of 1-forms with compact support cohomologous to φ, as in the proof of Proposition 2.7. Now

kαkLA = kφikLA = kφikL1 ≤ |φi|L1 ≤ |α|L1 + , (3.7) where → 0 as i → ∞. Thus we have

1 kαkLA ≤ kαkL1 for α ∈ H . (3.8) The above Lemma in the closed case [BD17, Lemma 3.1] is a manifestation of a much more general principle, that the Poincaréduality for the closed manifold is an isometry, when we put Lp and Lq norms on the dual vector spaces, where p and q are Hölder conjugates. This is [BK04, Remark 7.2 (1)], which we provide for the convenience for the readers: Lemma 3.7. Let M be closed n-dimensional Riemannian manifold. The Poincaréduality map

1 PD :(H (M, R), k · kp) → (Hn−1(M, R), k · kq) is an isometry.

In the case of Lemma 3.6, the above result shows that the least area norm on H2 is the L∞-norm.

4 L∞-norm, Main theorem and The Proof

4.1 Bounding L∞-norm by L2-norm To prove the right-hand side of Inequalities (1.5), we ﬁrst need an inequality between the L∞-norm and the L2-norm for the 1-forms. We start with several lemmas.

Lemma 4.1. If f : H3 → R is harmonic and B is a ball of radius r centered about p then 1 |dfp| ≤ kdfkL2(B) (4.1) pv(r) where

v(r) = 6π(r + 2rcsch2(r) − coth(r)(r2csch2(r) + 1)) (4.2)

This is [BD17, Lemma 4.3]. There are different definitions of a Margulis constant, and 3 we use the one from [CS12, 1.0.2]. Let Γ ≤ Isom+(H ) be a discrete and torsion-free 3 subgroup. For any γ ∈ Γ and any P ∈ H , define dP (γ) = dist(P, γ · P ). A Margulis constant µ is a number such that the following is true: for any P ∈ H3, if x, y ∈ Γ such that max(dP (x), dP (y)) < µ (4.3) then x and y commute. A Margulis constant gives rise to a thick-thin decomposition: M = Mthin ∪ Mthick where

18 Mthin = {m ∈ M|injm < µ/2} and Mthick = {m ∈ M|injm ≥ µ/2}.

We call Mthin the thin part and it is a disjoint union of Margulis tubes and rank-2 cusps. When H1(M) 6= 0, µ = 0.29 is a Margulis constant by [CS12]. We ﬁrst generalize [BD17, Theorem 4.1] to maximal cusp neighborhoods. Now ﬁx one of the maximal cusp neighborhoods, C0 and its boundary torus T0 := {z = 1}. By slightly shrinking the cusp 0 neighborhood√C , we obtain a smaller cusp neighborhood C whose boundary T is now based at z = 2. Lemma 4.2. Let M be a cusped hyperbolic 3-manifold, with a cusp neighborhood C and its boundary T described above. Denote the intrinsic diameter of T by d. Then we have for φ ∈ H1 r d2 kφk ∞ ≤ 2.43 1 + kφk 2 (4.4) L (C) 2 L (M) Proof. When acting on functions, the sign convention for the Laplace-de Rham operator ∆d and the Laplace-Beltrami operator ∆b is the following: ∆b = −∆d. For the proof of this lemma only, we reserve ∆ for the Laplace-Beltrami operator ∆b. Using the Bochner technique [Jos17, Theorem 4.5.1], we have

2 − ∆dhφ, φi = 2|∇φ| + 2Ric(φ, φ) (4.5)

If M instead had non-negative Ricci curvature, we would be able to conclude that

∆hφ, φi = −∆dhφ, φi ≥ 0 so that kφk∞ := kφkL∞(C) must occur on ∂C = T, by the properties of non-negative subharmonic functions vanishing at inﬁnity and a limit argument. To compensate for the negative curvature, we use the following identity: for a harmonic function f:

∆f 2 = 2|∇f|2 + 2f∆f = 2|∇f|2. (4.6)

Note that since φ is dual to a closed non-peripheral surface which can be homotoped away from the cusp, when restricted to C,

φ = df (4.7)

for some harmonic function f. The absolute value of a harmonic function is subharmonic. d In particular, from [MP90, (4.4)-(4.11)], combined with the identity dz zK1(z) = zK0(z) for the modiﬁed Bessel functions Kj(·), we have X φ = df, where f(x, y, z) = cizK1(λiz)ψi(x, y) (4.8) i=1

and ψi is the ith√ eigenfunction for ∆ of the boundary torus T with flat metric corre- 2 sponding to z = 2 with nonzero eigenvalue√ −λi . (For the full proof that φ = df, one can argue as in [BD17]: lift φ|C to {z ≥ 2} in upper half-space model. The expression (4.10) in [MP90√ ] is invariant under the corresponding parabolic isometries fixing infinity. Since {z ≥ 2} is simply connected, φe = df˜. We can pick f˜ to be the translation of

19 Equation (4.8).) Denote the restriction of f to T by f1. The collection of all eigenfunc- tions on T form an orthonormal basis for L2(T). Thus each non-constant eigenfunction ψi on T is orthogonal to the constant functions, which implies that Z f = 0 (4.9) T

It follows that the range of f1, contains 0 as an interior point: f1 cannot be identically non-positive or non-negative on T. Without this condition, it is impossible to bound f 2 by |df|max. We can now pick a constant c and swap f by

g := f + c (4.10) so that g1 := g|T = (f + c)|T satisﬁes

(g1)max = −(g1)min. (4.11)

2 2 This constant c also minimizes maxT g = max g1 among all such choices. Since 0 is an interior point of the range of g1, we have c < (g1)max. Now we can bound kφk∞ by kφ|Tk∞ and g|T. Rewriting Equation (4.5) using Equation (4.7), we have 1 ∆ |φ|2 = |∇2f|2 − 2|∇f|2 (4.12) 2 Combining with Equation (4.6), we have 1 ∆( |φ|2 + (f + c)2) = |∇2f|2 ≥ 0 (4.13) 2

1 2 2 and thus h := 2 |φ| + (f + c) is a subharmonic function, for any constant c. As z = t → ∞, f and |df| approach 0. Thus h(z) → c2. Consider C√t , the portion of the cusp √ 2 neighborhood from z = 2 to z = t, with two boundary components: T and Tt. Since h is subharmonic in C, its maximal is achieved either on T or on Tt. By choosing a large enough t, we can arrange that max h < max h (4.14) Tt T It then follows that max h = max h (4.15) C T Now we have

2 1 2 2 kφk∞ ≤ 2 max( |φ| + g ) C 2 1 = 2 max( |φ|2 + g2) T 2 ≤ max |φ|2 + 2 max g2 T T

20 The c we choose implies g − g max g2 = ( max min )2 (4.16) T 2

where gmax and gmin are the maximum and the minimum of g restricted to T, respectively. Using the mean value theorem, we have on T

gmax − gmin ≤ d · |dg|max (4.17)

Here we also use the fact that the torus T is a submanifold and |dg|T ≤ |dg|hyp pointwisely. Thus we have 2 2 d 2 kφk∞ ≤ (1 + ) max |φ| (4.18) 2 T

Using the lower bound of waist size from [Ada02], we have injT is larger than√ 0.48. The largest ball embedded in C0 is centered at T and has hyperbolic radius ln 2. By Lemma 4.1, we have 1 max |φ| ≤ q √ kφkL2(M) ≈ 2.43kφkL2(M) (4.19) T ν(ln( 2))

where ν(r) is deﬁned in Equation (4.2).

Now we have Proposition 4.3. For an orientable cusped hyperbolic 3-manifold M, we have

r 2 5 d 1 k · kL∞ ≤ max{ , 2.43 1 + }k · kL2 on H , (4.20) psys(M) 2

where d is the largest diameter of the boundary tori of the maximal cusp neighborhoods.

1 Proof. Assume H 6= 0. For µ = 0.292, each cusp component of Mthin is contained in a 1 maximal cusp to which Lemma 4.2 applies. For φ ∈ H , if kφkL∞ is not realized in the 5 cusp neighborhoods, then kφkL∞ ≤ √ kφkL2 by [BD17, Theorem 4.1]. sys(M)

4.2 Main theorem and the proofs Now we are ready to prove Theorem 1.5 For all cusped orientable hyperbolic 3-manifolds M one has

r 2 π 10π d 1 k · kT h ≤ k · kL2 ≤ max{ , 4.86π 1 + }k · kT h on H , pvol(M) psys(M) 2 where d is the largest diameter of the boundary tori of the maximal cusp neighborhoods. Here we present a proof whose argument is similar to [BD17] using local estimates and properties of minimal surface in hyperbolic 3-manifolds.

Proof. Let α ∈ H1. For the left-hand inequality, it is a simple application of Cauchy- Schwarz inequality. By Corollary 3.5 and Equation (3.8) we have

21 p πkαkT h ≤ kαkLA ≤ |α|L1 = ||α| · 1|L1 ≤ kαkL2 k1kL2 = kαkL2 vol(M).

Thus we have π kαkT h ≤ kαkL2 . pvol(M)

Now we prove the right-hand inequality. By continuity of the norms, it suffices to prove for the integral coefficients. For the closed case, the remaining argument is the following. Fix a closed surface S dual to a harmonic 1-form α, of area at most 2πkαkT h [BD17, R R Theorem 3.2]. By Poincaréduality that M β ∧ φ = S β for every closed 2-form β. In the closed case, the harmonic form φ is closed and coclosed. Thus we have Z Z Z 2 kαkL2 = α ∧ ∗α = ∗α ∧ α = ∗α M M S Z Z Z ≤ | ∗ α|dA = |α|dA ≤ kαkL∞ dA S S S ≤ kαkL∞ Area(S) ≤ 2πkαkL∞ kαkT h, where we use Proposition 3.3. Using Equation (2.14) we have Z 2 1 kαkL2 = ∗α for α ∈ H . S

The remaining part of the proof is similar to the closed case in [BD17]. Note that we ∞ only need to bound the L -norm of α on the surface S ⊂ Mτ .

The proof of Corollary 1.7 follows naturally: Corollary 1.7 Let V0 be a ﬁxed positive constant. For all but ﬁnitely many cusped manifolds M with volume less than V0, we have 10π k · kL2 ≤ k · kT h (4.21) psys(M)

Proof. As the waist size is at least 1, if the diameter d of a cusp tends to infinity, then the volume of the corresponding cusp goes to infinity. It implies that the diameter d is bounded in terms of V0. On the other hand, by the Thurston-Jørgensen theory, the number of hyperbolic 3-manifolds with volume ≤ V0 and systole ≥ 0 is finite, where 0 > 0 is a constant. Thus for all but finitely many cusped manifolds M with volume ≤ V0, the systole term in the right-hand side of Inequalities 1.5 dominates the diameter term.

5 Sharpness of the Inequalities and a Functional Point of View

In this section we assume the manifold is closed as in Inequalities (1.3), unless noted otherwise. We only study the left-hand side. The right-hand side for both closed and cusped M is not sharp and will be investigated in the following papers.

22 5.1 Three conditions for the sharpness and their interactions Proposition 5.1. If α ∈ H1 is a harmonic 1-form achieving equality in π k · kT h ≤ k · kL2 , pvol(M) it must satisfy the following three conditions:

1. It has constant length: |α|p = c on M. 1 R 1 1 2. The L -norm of α, |α|L1 is given by M |α|. If φ ∈ H corresponds to α, the L - R 0 norm on its cohomology class is kφkL1 = infφ0∼φ∈H1 M |φ |. The second condition is that |α|L1 = kφkL1 (5.1)

3. Let S be a least area taut surface dual to α. The last condition is that

Area(S) = πχ−(S) (5.2)

R 2 R 2 2 R It is equivalent to S |σ| = 2Area(S) or S h11 + h12 = S 1, where σ is the second fundamental form associated to S with components h11, h12, h21, h22. The proof of the above proposition is just bookkeeping, see the proof of [BD17, Theorem 1.2, 3.2]. Condition1 is more likely a phenomenon on a Seifert ﬁbered manifold than hyperbolic ones, for example, dz on 3-torus T3 glued from unit cube in the Euclidean space. Condition2 also seems unlikely as commented in [BD17]: kφkL1 is typically realized by a non-smooth form, supported like a Dirac measure around the surface. Condition3 is not possible and discussed in Proposition 5.4. Our ﬁrst observation is that, surprisingly, Condition1 implies2 and3. Proposition 5.2. Let M be a closed orientable hyperbolic 3-manifold. If α is a nontrivial L2 harmonic 1-form with constant length, dual to a closed surface S, then

Area(S) = πχ−(S)

Proof. Let |α|p = c, where c is a constant. By the Bochner formula for 1-forms, we have

1 2 2 2 2 2 2 2 ∆(|α| ) = |∇α| − 2|α| =⇒ |∇α| = 2|α| = 2c

As a closed form, locally in a ball α = df. Denote ∇f by V . Since |V | = |α| is constant, we have for all vector ﬁelds W on M,

0 = h∇W V,V i = Hess(f)(W, V )

The Hessian of f Hess(f) = {hij} is a symmetric 3 × 3 matrix, with ﬁrst two columns corresponding to tangent vectors to S, and last column to V . The above equation implies that the last column and the last row consist of 0’s. Thus we have |∇α| = |∇α|S|. The second fundamental form on S is ∇α σS = (5.3) |α| S which implies that

23 Z Z Z 2 2 2 1 2 |σS| = 2 =⇒ h11 + h12 = |σS| = 1 S 2 S S

The rest of the arguments are identical to the proof of [Has95, Lemma 6]. Using the Gauss-Bonnet theorem, the Gauss formula and the fact that the metric is hyperbolic, we have

R 2 2 R S h11 + h12 = −2πχ(S) + S −1 = −Area(S) − 2πχ(S)

which gives Area(S) = −πχ(S) = πχ−(S).

The proof above is partially inspired by the proof of [Ste19, Theorem 1.1]. Now we prove that Condition1 implies2 on a general Riemannian manifold. This is a straightforward 1 corollary of [Kat07, Proposition 16.9.1]. There whether k · kL1 on H is realized by a smooth form is not investigated. Proposition 5.3. Let M be a closed Riemannian n-manifold. Let α be a nontrivial harmonic 1-form with constant length. Then it realizes the L1-norm in its cohomology class.

Proof. From [Kat07, Proposition 16.9.1], we have that existence of a nontrivial harmonic 1-form α with constant length implies that 1 1 kαkL1 = kαkL2 (5.4) vol(M) pvol(M)

The 1-form α has constant length c implies that q R R 2 p |α|L1 = M |α| = c · vol(M) and kαkL2 = M |α| = c vol(M)

Thus kαkL1 = c · vol(M) = |α|L1 .

In contrast to the closed hyperbolic 3-manifolds, in the 3-torus, the incompressible 2- torus as in Figure2 gives rises to a dual harmonic 1-form dz which realizes its least area and hence its L1-norm.

5.2 Proof of non-sharpness using harmonic 1-forms and minimal foliations Now we verify our intuition about constant length harmonic 1-forms: such forms should not be expected in hyperbolic manifolds. Manifolds all of whose harmonic (1-)forms are interesting in their own right, and have connections with systolic geometry, see [NV04] and citations therein. Proposition 5.4. Let M be a closed orientable hyperbolic 3-manifold. Any nontrivial harmonic 1-forms does not have constant length.

24 Proof. Using [MR17, 8.1] and [G09´ , Theorem 12], an immersed stable orientable closed minimal surface S of genus g in a closed orientable hyperbolic 3-manifolds cannot have area 2π(g −1), contradicting Proposition 5.2. One can look at the zeros of a holomorphic quadratic diﬀerential for a slightly more direct argument for non-existence of harmonic 1-forms with constant length. By the proof of Proposition 5.2, the existence of such a 2 1-form implies that the second fundamental form σS satisﬁes |σS| = 2 on the dual surface S. By [Kat07, Lemma 16.7.1], S is minimal. Minimality of S and the fact that M is constant sectional curvature imply that

2 σS = Re(h11 − ih12)(dz) = Reη (5.5)

2 where η = (h11 − ih12)(dz) is a invariantly defined holomorphic quadratic differential. Any such differential must vanish at exactly 4g − 4 ≥ 4 points, where g = g(S). Thus it 2 is impossible that |σS| = 2 identically.

The introduction of the notion of a geometric foliation and its non-existence in closed hyperbolic 3-manifolds [WW20, Theorem 1.2] provides yet another point of view. We first need a lemma which relates harmonic 1-forms with constant length to minimal foliations [Lemma 16.7.1, Kat07]: Lemma 5.5. If α is a harmonic 1-form of constant length, then the leaves of the distri- bution ker(α) are minimal surfaces, and the vector field V that is dual to α has geodesic flow lines orthogonal to ker(α). Note that Zeghib [Zeg93] proved that there is no smooth vector field on closed hyperbolic 3-manifold where all the flow lines are geodesic (the flow lines given by harmonic 1-forms are smooth, while Zeghib only require continuity). The foliation described above is a specific example of geometric foliation defined in [WW20, Definition 1.4]: Definition 5.6. Let S be a closed surface, > 0 a constant, h an embedding

h :(−, ) × S → M (5.6)

We say h is a geometric 1-parameter family of closed minimal surfaces if 1. h is C2 with respect to both t and p ∈ S.

2. ∀t, the leaf ht ⊂ M is a minimal surface.

3. ∀p ∈ S, f(t, p) := h(ht)∗(∂t), νi|t=0 only depends on the principal curvature of S at 2 p. One can write f(0, p) = f(0, kσSk (p)) where σS is the second fundamental form of {0} × S at (0, p) in M.

4. f(0, ·): S → R is not identically 0. Note that a harmonic 1-form α gives rise to a foliation by Lemma 5.5 with geodesic ﬂow line orthogonal to each leaf. This is the strongest geometric foliation one can hope for in Deﬁnition 5.6 in which f ≡ 1. Now [WW20, Theorem 1.4] states that in a closed hyperbolic 3-manifold there does not exist a geometric foliation by minimal surfaces, and hence no harmonic 1-forms can have constant length. The proof of non-sharpness is a simple corollary.

25 Corollary 5.7. There does not exist a closed orientable hyperbolic 3-manifold M such that

π 1 kαkT h = kαkL2 for some α ∈ H pvol(M) The cusped case is, surprisingly, easier than the closed case, due to the asymptotic estimate near inﬁnity. To achieve equality, the same arguments imply that we still need to satisfy the three conditions in 5.1. The second condition α has constant operator norm is no longer possible, as it decays exponentially near the cusp. Corollary 5.8. There does not exist an orientable cusped hyperbolic 3-manifold M such that

π 1 kαkT h = kαkL2 for some α ∈ H pvol(M)

5.3 A functional point of view

In this section, we define two functionals Di and Ds on the collection Ψ := {Closed or cusped orientable hyperbolic 3-manifolds}. It unifies various phenomena in the last section and in [BD17] and allow us to ask new interesting questions. The definitions are

πkαkT h Di(M) := inf (5.7) 1 p α∈H vol(M)kαkL2 and πkαkT h Ds(M) := sup p (5.8) α∈H1 vol(M)kαkL2 The left-hand side of Inequalities (1.5) and the rigidity discussed in the last section can be phrased in terms of Di ≤ Ds < 1. Using the right-hand side of Inequalities (1.3) or Inequalities (1.5), we have k · k pinj(M) inj(M ) T h ≥ or τ k · kL2 10π 8π

for closed and cusped M, respectively. Thus 0 < Di ≤ Ds. Summarizing both sides, we have 0 < Di ≤ Ds < 1 on Ψ (5.9) Which functional is stronger in a statement depends on whether we are interested in behaviors near 1 or near 0. Note that when b1 = 1, Di = Ds. We simply use D to refer to either Di or Ds when it is not very important to choose one over the other. Using an algorithm, Dunfield and Hirani [HD20] found examples of fibered hyperbolic 3-manifolds where D = 0.95 and D = 0.994, which is very surprising (not every fibered manifold satisfies this property). The rigidity is realized by a product metric in [BS19], so it is natural to conjecture that

Conjecture 5.9. There exists an > 0 such that {M|Di(M) > 1−}={ﬁbered hyperbolic 3-manifolds with some special geometry} and that there exists a sequence of closed Mi such that Di(Mi) → 1.

26 We are also interested in the behavior of D(Mi) < , where is a small positive number. We ﬁrst incorporate a theorem in [BD17] into a behavior of the functional.

1 Theorem 5.10. ([BD17], Theorem 1.4) There exists a sequence of Mn and φn ∈ H (Mn; R) so that

1. The volumes of the Mn are uniformly bounded and inj(Mn) → 0 as n → ∞. p 2. kφnkL2 /kφnkT h → ∞ like − log(inj(Mn)) as n → ∞.

The Mn above comes from the Dehn ﬁlling on the complement of the link L = L14n21792, all with Betti number 1. It can be rephrased in terms of D:

Corollary 5.11. There exists a sequence of Mn from Dehn ﬁlling the complement of a link so that

D(Mn) → 0

One can ask how to classify all sequences of Mn such that D(Mn) → 0. One could take ﬁnite covers, as the following theorem shows that taking ﬁnite cover bound Di from above:

1 Theorem 5.12. ([BD17], Theorem 1.3) There exists a sequence of Mn and φn ∈ H (Mn; R) so that

1. The quantities vol(Mn) and inj(Mn) → ∞ as n → ∞.

πkφnkT h 2. The ratio p is constant. vol(Mn)kφnkL2 The proof uses the fact√ that the lift of a harmonic representative is also harmonic, and ∗ hence kπ (φ)kL2 = dkφkL2 , where d is the degree of a ﬁxed cover π : Mf → M. The Thurston norm scales linearly, by a deep theorem of Gabai [Gab83, Corollary 6.13]: ∗ kπ (φ)kT h = dkφkT h. Thus we have

Corollary 5.13. Let Mf be a ﬁnite cover of M. Then

Di(Mf) ≤ Di(M) ≤ Ds(M) ≤ Ds(Mf) (5.10)

Due to the structure of Ψ, we expect most values of D to be discrete, with accumulation points only at cusped manifolds corresponding to geometric convergence.

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