Heat Flow Methods in Geometry and Topology

H. Dietert, K.Moore, P. Rockstroh, G. Shaw April 1, 2014

Introduction

n R 2 For a domain Ω ⊂ R , the Dirichlet energy functional Ω |∇f| dx, defined for functions f ∈ C1(Ω, R), is a central object of interest in analysis and has been widely studied. Its minimisers satisfy the Laplace equation ∆ f = 0 and this characterisation shows that the minimisers enjoy many special properties, e.g. infinite differentiability. We can generalise the Dirichlet energy functional E(f) to functions f ∈ C1(M, N ) between abstract Riemannian manifolds (M, g) and (N , h). The stationary points satisfy the Euler-Lagrange equation and are the generalisation of harmonic maps. The first systematic treatment of harmonic maps between manifolds of arbitrary dimension was undertaken by James Eells and J.H. Sampson in their seminal paper, ‘Harmonic Mappings of Riemannian Manifolds’ [3]. The basic question they tackled was if given a function f ∈ C0(M, N ), there exists a harmonic function which is homotopic. The main idea is to study the studying a corresponding heat flow on the manifold under which E(f) must decrease, and by analysing the long time behaviour of solutions, to find a related harmonic map. They proved that, in the case that the target manifold N has non-positive sectional curvature, a solution to the heat flow equation exists for all time, and further, as t → ∞, a harmonic map occurs in the limit. This report gives a self-contained discussion of these results and in particular gives a rigorous complete treatment for the existence of the heat flow in a very general setting. Our exposition is mostly a combination of the presentation in Eells and Sampson’s original paper ([3]) and the presentation in

1 Lin and Wang’s book ‘Harmonic Maps and Their Heat Flows‘, [6]. The organisation of the report is as follows: Section 1 provides a brief overview of necessary prerequisites from differential geometry and Section 2 introduces the Dirichlet integral for maps between manifolds M and N and defines harmonic maps as the critical points of this functional. It is also shown that harmonic maps are characterised as solutions to the tension field, the system of Euler-Lagrange equations associated to the Dirichlet integral. Section 3 outlines a heuristic strategy for obtaining a harmonic map as the steady state solution to the heat flow associated with the tension field. This heat flow is analysed by using the Nash Embedding Theorem to embed the tension field into Euclidean space as a system of equations valid for maps taking values in RL in Section 4. Next, in Section 5 the theory for the linear heat equation, valid for scalar valued maps defined on M, is developed. In Section 6, the results of Sections 4 and 5 are combined with a fixed point argument to prove that the heat flow must always admit a local solution and to derive a blow up criterion for the non-existence of a global solution. Section 7 completes the report by showing that, if the target manifold N is assumed to have non-positive sectional curvature, no such blow up can occur and by then using the existence of a global solution for the heat flow to construct the required homotopy.

2 Contents

1 Tools from diﬀerential geometry 4

2 Harmonic maps and the energy functional 6

3 The heat ﬂow 8

4 Global equations 9

5 The heat kernel an a Riemannian manifold 14 5.1 Deﬁnition Of The Heat Kernel And Its Properties ...... 14 5.2 Construction of the Heat Kernel ...... 19

6 Short time existence 28

7 Global solution to the heat ﬂow 33 7.1 A priori estimates ...... 33 7.2 Convergence at inﬁnity to a harmonic map ...... 39

8 Conclusion 39

3 1 Tools from diﬀerential geometry

A smooth m-dimensional manifold M looks, at every point P ∈ M, like Rm, and the relation is smooth. That is, for all P ∈ M, there is a smooth diffeomorphism ϕ between a neighbourhood of P and an open subset of Rm. We refer to the Euclidean coordinates of ϕ(P ) as local coordinates for P in M and denote them as x1, x2, . . . , xm and this coordinate system extends to the neighbourhood of P on which ϕ is defined. The idea of tangent space to M at P is intuitively easy to imagine. You have come across the tangent to a curve; in this case the tangent space is just the line tangent to the curve at that point. Now picture a sphere. Pick a point on the sphere and imagine what a tangent vector would look like. Can you see that the tangent space is this time a plane? We may have a tangent vector field on a manifold, which sends each point of M to a vector in its tangent space. We may express a tangent vector field X in terms of ∂ ∂ ∂ the basis ∂x1 , ∂x2 ,..., ∂xm as ∂ X = Xi , (1.1) ∂xi where we use the summation convention. However, these coordinates are only valid locally, so if we have a sequence of maps into a manifold, it is not necessarily possible to describe the image of a small region in the domain with one system in local coordinates. A metric g introduces length and angles on the manifold by describing a inner product between two tangent vectors, i.e.

g : T M × T M → R. (1.2)

With respect to a local coordinate system we may write g|p as a matrix, if we plug in the basis vectors to the tangent space at a point p, i.e. ∂ ∂ g := , ij ∂xi ∂xj

ij We also denote by g the inverse matrix to gij. A smooth manifold equipped with a metric, (M, g), is known as a Riemannian manifold. Notice that the inverse also induces an inner product on cotangent vector fields, which are dual to tangent vector fields. With this we can define the distance of two points as the length of the shortest path. As the manifold behaves locally

4 like Rm, the shortest path is unique for points sufficiently close to each other. The radius of the largest ball within which this distance is well defined is called the radius of injectivity. At any point P ∈ M we can construct local normal coordinates xi using an object called the exponential map, which is constructed so that at P the metric looks as close to the Euclidean metric as possible. These coordinates are especially useful for calculations and we will make use of them. If we have a vector field X and local coordinates xi, we can define the partial derivative as ∂X ∂Xj ∂ X := = . (1.3) ,i ∂xi ∂xi ∂xj However, the resulting vector is not coordinate independent. A connection ∇ is a coordinate independent operator that behaves like a derivative, i.e. obeys the Leibniz rule. For a given metric g there exists the unique Levi-Civita connection (which we refer to as the connection from herein), which is torsion free, i.e. ∇i∇jf = ∇j∇if for a scalar function f, and is compatible with the metric, that is, ∇igjk = 0. We always use this connection and denote X;j = ∇jX. (1.4) In a coordinate basis this takes the form ∂Xi Xi = + Γi Xk, (1.5) ;j ∂xj jk i where Γjk are the Christoffel symbols which, for the Levi-Civita connection are 1 Γi = gil(g + g − g ). (1.6) jk 2 jl,k kl,j jk,l In normal coordinates centered at P , the Christoffel symbols vanish and the covariant derivative and partial derivative agree when all are evaluated at P . If we want to integrate over the manifold we need to find a coordinate independent way. On a Riemannian manifold, this can with local coordinate functions and integration behaves as we would expect. Furthermore, the divergence theorem with covariant derivatives holds. When we express this in the coordinate system xi which covers the sup- port of f and maps to V ⊂ Rm, the integral becomes Z Z √ f(P )dVg(P ) = f(ϕ(x)) gdx M V

5 √ where g is the determinant of the matrix gij, i.e. g is the volume element associated to the metric g. We introduce the Laplace-Baltrami operator ∆g as geometric operator (i.e. coordinate independent) which reduces in Euclidean case to the normal Laplace operator. For this we deﬁne

ij ∆g f = g f;ij. (1.7)

In the coordinate system xi, this can be equivalently expressed as

m √ X ∂i( g∂if) ∆ f = √ g g i=1 2 Harmonic maps and the energy functional

Consider a smooth map f :(M, g) 7→ (N , h) between smooth Riemannian manifolds (M, g) and (N , h) where both manifolds are without boundary and compact. Throughout the report we will assume compactness and use it at various places. We denote coordinates on M with the indices i, j, . . . and on N with the indices α, β, . . .. Definition (Energy Density). For f ∈ C1(M, N ), the energy density e(f) is defined to be 2 e(f) := |df|g , (2.1) where df is the differential of f, or in local coordinates centred at a point P , 1 e(f)(P ) = gij(P )h (f(P ))f α f β . (2.2) 2 αβ ,i ,j

If f is a map f : Rk 7→ Rd, then df is the Jacobian matrix of f and e(f) is just a multiple of the Frobenius norm of df. Deﬁnition (Energy Functional). For f ∈ C1(M, N ), the energy of f is deﬁned as Z E(f) = e(f)dVg. (2.3) M So E(f) is a generalisation of the Dirichlet energy functional to maps between manifolds. Note that, since e(f) is non-negative, E(f) is as well.

6 Definition (Harmonic Maps). A map f ∈ C2(M, N ) is said to be harmonic d if f is a critical point of E(·), that is, dt E(ft) t=0 = 0 for any curve ft : 2 (−, ) → C (M, N ) satisfying f0 = f. The question that we are interested in answering is, given a homotopy class [f] ⊂ C(M, N ), does there exist a harmonic representative of [f]? Equivalently, can every f ∈ C2(M, N ) be continuously deformed into a harmonic map? Definition (The Tension Field). Define the tension field τ : C2(M, N ) → T (N ) by Z d dft E(ft) = − hτ(f), idVg (2.4) dt t=0 M dt t=0 2 where (ft) ∈ C (M, N ) is a family of pertubations of f in C (M, N ) such that f0 = f as before. The field τ is the Euler-Lagrange operator associated to E. We note that f ∈ C2(M, N ) is harmonic if and only if τ(f) = 0. Lemma 2.1. For f ∈ C2(M, N ) the tension field τ is given by

α α ij N α β γ τ(f) (P ) = ∆g f (P ) + g (P )(Γ ) βγ(f(P ))f ,if ,j. (2.5)

2 Proof. For f ∈ C (M, N ) let (ft)t∈(−,) be a smooth one parameter family d of functions such that f0 = f and ϕ = dt ft t=0. We then find Z d 1 d ij α β [E(ft)]t=0 = g (P )hαβ(ft(P ))ft ,ift ,jdVg dt 2 dt M t=0 Z ij α β 1 α β γ = g (hαβft ,i∂tft ,j + hαβ,γft ,ift ,j∂tft )dVg. M 2 We can define a tangent field ξ on M by

i ij α β ξ = g ft ,jϕ hαβ, (2.6) α α α where ft ,j = ft ;j as ft is a scalar function with respect to M. By Green’s divergence theorem Z i 0 = ξ ;idVg M Z (2.7) ij α β α β α β γ = g (ft ;jiϕ hαβ + ft ,iϕ ,jhαβ + ft ,jϕ hαβ,γft ,i)dVg. M

7 Plugging this back gives Z d ij α β α β γ 1 α β γ [E(ft)]t=0 = g (−ft;jiϕ hαβ −ft,jϕ hαβ,γf , t,i + hαβ,γft ,ift ,jϕ )dVg. dt M 2 (2.8) The ﬁrst term we recognise as Laplace-Baltrami operator ∆g deﬁned as

α ij α ∆g ft = g ft ;ij. (2.9) For the remaining term we ﬁnd using the symmetry of h 1 1 f α ϕβh f γ − h f α f β ϕγ = ϕαf β f γ (h + h − h ) t ,j αβ,γ t ,i 2 αβ,γ t ,i t ,j t ,i t ,j 2 βα,γ γα,β βγ,α | {z } N δ hαδ(Γ )βγ (2.10) where we recognise the Christoﬀel symbol ΓN .

3 The heat ﬂow

Our goal is, for a given g ∈ C(M, N ), to a find a map f ∈ [g] such that E(f) is a critical point. If ft satisfies the heat flow ( ∂tft = τ(ft)

f0 = g then we have that Z Z d 2 (E(ft)) = − hτ(ft), ∂tftidVg = − |∂tft| dVg. (3.1) dt M M

If we can prove that ft is defined under this heat flow for all time and that there is some kind of convergence ft → f∞ as t → ∞, then, since E(f)∞ is bounded below by 0, we would expect that ∂tf∞ = limt→∞ ∂tft = 0 and hence that τ(f∞) = 0. Since f∞ is homotopic to f0 via the heat flow, we would then have that f∞ is a harmonic representative of [g]. Morally, this is the strategy which we pursue, albeit in a slightly disguised form. In order to show the existence of a solution, we must consider the flow embedded in a Euclidean space RL. We will prove that this embedded flow admits a unique local solution and then prove that, if the curvature of N

8 is non-positive, this solution is global. We will then prove that this global solution admits a subsequence converging to a harmonic limit f∞ and that this subsequence is eventually ‘close enough’ to f∞, at which point a geodesic ﬂow can be used to complete the homotopy.

4 Global equations

The equation ∂tf = τ(ft) is a system of semilinear PDEs in terms of the local coordinates of N . To avoid problems with local coordinates and to allow us to prove the existence of a solution, we analyse the heat flow ∂tft = τ(ft) by embedding the manifold N into some Euclidean space RL. We then prove L that a solution ft ∈ C(M, R ) exists and that ft(M) ⊂ N for all time separately. In order to do this, we will need to reformulate ∂tft = τ(ft) in terms of equations involve coordinates in RL as opposed to coordinates in N . By the Nash Embedding Theorem, there exists L ∈ N and w : N 7→ RL such that w is an isometric embedding of N into RL with the Euclidean metric. Define u = w ◦ f, the embedding of f into RL. In order to describe τ(f) in RL, we must find an explicit expression for dw ◦ τ(f), the embedding of the vector field τ(f) into RL, in terms of u. By compactness for small enough δ0 > 0, the nearest point projection map π : Nδ0 → N , where

L Nδ := {y ∈ : d(y, N ) := inf |z − y| < δ0}, (4.1) 0 R z∈N is unique and smooth. We will always use standard Euclidean coordinates in RL (and hence

Nδ0 ) whose components are denoted by the indices a, b, . . .. In Euclidean coordinates the metric then reduces to the Kronecker delta δ.

Lemma 4.1. The embedded tension ﬁeld dw ◦ τ(f) is the projection of ∆g u onto T (N ), ie dw ◦ τ(f) = dπ ◦ ∆g u. (4.2) Proof. This is a pointwise equality and so we prove it separately for every point P ∈ M. We choose local normal coordinates for M and N centered at P and f(P ). In these coordinates the Christoﬀel symbols vanish so that

a ij a ij a ∆g u (P ) = g u ;ij(P ) = g u ,ij(P ) (4.3)

9 and ij τ(f)(P ) = ∆g f(P ) = g f,ij(P ). (4.4) By the chain rule we find a a a α a α β u ,ij = (u ◦ f) ,ij = w ,αf ,ij + w ,αβf ,if ,j. (4.5) At P , this gives a ij a α ij a α β ∆g u (P ) = g w α(f(P ))f ,ij(P ) + g w ,αβ(f(P ))f ,i(P )f ,j(P ) a α ij a α β (4.6) = w α(f(P ))τ(f) (P ) + g w ,αβ(f(P ))f ,i(P )f ,j(P ). The first term is (dw ◦ τ(f))a, so, putting this all together, we have ij α β ∆g(u) = dw ◦ τ(f) + g w,αβ(f(P ))f ,i(P )f ,j(P ). (4.7) ij α β It remains to show that g w,αβ(f(P ))f ,i(P )f ,j(P ) is normal to T (N ). To see this, let xα be the system of normal coordinates centered at f(P ) in N and let X,Y be two vector fields defined in a neighbourhood of f(P ) α α ∂ whose coefficients X ,Y with respect to the basis ∂xα are constant. Then, at f(P ), α β α β hX,Y i,γ = (hαβX Y ),γ = hαβ,γX Y = 0 (4.8) since the first partial derivatives of h within these coordinates are zero at f(P ). Since w is an isometric embedding, we have a b α β 0 = hdw ◦ X, dw ◦ Y i,γ = (δabw,αw,βX Y ),γ a b a b α β (4.9) = δab(w ,αγw ,β + w ,αw ,βγ)X Y . As X and Y are arbitrary vectors, the bracket itself must vanish. Permuting the indices we find three equations a b a b 0 = δab(w ,αγw ,β + w ,αw,βγ), a b a b 0 = δab(w ,γβw ,α + w ,γw,αβ), a b a b 0 = δab(w ,βαw ,γ + w ,βw,γα). Now add the first two equations and subtract the third one. Using the symmetry of δab the first term in the first equation is cancelled by the second one in the third one. Likewise the second term in the second equation is cancelled by the first term in the third one. Finally the second term in the first one and the first term in the second equation are equal, so that we find a b 0 = δabw ,αw ,βγ = hw,α, w,βγi. (4.10) a As w ,α spans the tangent space, this shows orthogonality.

10 Next, we consider the decomposition of ∆g u into its parts parallel and normal to TN in another way. Using local coordinates again,

a a a ij a b ij ∆g u = ∆g π(u) = (π(u) ),ijg = (π ,bu,i),jg a b c a b ij = (π ,bcu,iu,j + π,bu,ij)g a a b c ij = dπ ◦ (∆g u) + π ,bcui ujg a a α β ij a b c ij = (dπ ◦ dw ◦ τ(f)) + dπ ◦ w,αβf,i f,j g + π ,bcui ujg a a b c = dw ◦ τ(f) + π ,bcui uj,

α β ij since dπ is the orthogonal projection onto TN and w,αβf,i f,j g is normal to T (N ). Finally, then, we have an expression for τ(f) in Euclidean coordinates:

dw ◦ τ(f) = ∆g u + A(u)(∇u)(∇u) (4.11) where A(u)(∇u)(∇u)1 is deﬁned by

a ij a b c A (u)(∇u)(∇u) = −g π ;bcu ,iu ,j. (4.12)

a a a α By the chain rule, ∂tut = ∂t(w (ft)) = wα∂tft = dw ◦ ∂tft, so ut = w ◦ ft satisfies the heat flow ( ∂ u − ∆ u = A(u )(∇u , ∇u ) t t g t t t t (4.13) u0 = w ◦ g whenever f satisfies t ( ∂ f = τ(f ) t t t (4.14) f0 = g.

Theorem 4.2. If ut is a solution of the heat ﬂow 4.13 for some initial condition w ◦ g where g ∈ C(M, N ) then ut(M) ⊂ N for all t such that ut exists.

L Proof. Deﬁne the deviation from N as ρ : Nδ0 → R by ρ(x) = x − π(x). Taking the derivative of the deﬁnition shows

a a a a a ρ ,b + π ,b = δb and ρ ,bc + π ,bc = 0. (4.15)

1This can be further identiﬁed as the second fundamental form. We only need that this is a quadratic form in ∇u.

11 We will show that ρ(ut) = 0 for all t for which ut is deﬁned. First, we ﬁx normal coordinates about a point P in order to calculate (∂t − ∆g)ρ(ut) at P .

a a c c ij a b c (∂t − ∆g)ρ(ut) = ρ ,c (∂tut − ∆g ut ) − g ρ ,bcut ,iut ,j a ij c d e ij a b c = −ρ ,cg π ,deut ,iut ,j + g π ,bcut ,iut ,j a a ij c d e = (−ρ ,c + δc )g π ,deut ,iut ,j a ij c d e = π ,cg π ,deut ,iut ,j.

From which we deduce (∂t −∆g)ρ(ut) = (dπ◦A)(ut)(∇ut, ∇ut). This enables us to compute what happens to the modulus of ρ under the heat operator, 2 a b (∂t − ∆g)(|ρ(ut)| ) = (∂t − ∆g) δabρ (ut)ρ (ut) a b b = δabρ (ut) ∂tρ (ut) − ∆g ρ (ut) b a a + δabρ (ut)(∂tρ (ut) − ∆g ρ (ut)) ij a b b a − δabg ρ (ut),iρ (ut),j + .ρ (ut),iρ (ut),j 2 = 2hρ(ut), (∂t − ∆g)ρ(ut)i − 2|∇ρ(ut)| 2 = 2hρ(ut), (dπ ◦ A)(ut)(∇ut, ∇ut)i − 2|∇ρ(ut)| 2 = −2|∇ρ(ut)| since (dπ ◦ A)(ut)(∇ut, ∇ut) takes values in T (N ) and ρ(ut), as the vector between ut and its nearest point on N , is perpendicular to T (N ). Since 2 ρ(u0) = 0, we can appeal to the maximum principle to deduce that |ρ(ut)| ≤ 0 for all t such that ut is deﬁned. Alternatively, we can note that we have just shown that

2 2 ∂t|ρ(ut)| ≤ ∆g |ρ(ut)| . (4.16)

Integrating across M and using the divergence theorem on ∆g = div ◦ grad yields Z Z 2 2 ∂t |ρ(ut)| dVg ≤ − |∇ρ(ut)| dVg ≤ 0. (4.17) M M

Hence, for initial data in C(M, N ), the flow under 4.13 is equivalent to the flow under 4.14. In particular, any limit u = limk→∞ utk of a solution ut to 4.13 satisfying ∆g u + A(u)(∇u)(∇u) = 0 gives rise to a harmonic map f = w−1 ◦ u and so we are justified in focusing our attention on solving 4.13.

12 Remark. It is worth noting that, if we define E˜ as the Dirichlet energy for 1 L ˜ maps in C (M, R ), the operator ∆g is the tension fieldτ ˜ for E. Since w is an isometric embedding, u = w◦f, where f ∈ C1(M, N ), satisfies E˜(u) = E(f). ˜ Naively, we might expect to be able to useτ ˜ = ∆g to decrease E(u) and hence E(f) and to obtain our harmonic map in the limit. However, all maps with image in RL are homotopic to constant maps, so, given convergence in the limit, the heat flow operator ∂t − ∆g will contract any initial data w ◦ g to a single point, and, in our case, forcing the image of our initial data g to leave N . The term A(u)(∇u, ∇u), which we recall is normal to the surface of N , can be seen as a force which corrects the action of ∂t − ∆g by ‘pushing back’ against it to ensure that our solution always stays on the manifold N by negating the normal component of the flow, leaving only tangential forces which flow the image of g within N .

13 5 The heat kernel an a Riemannian manifold

In order to construct the heat flow between the manifolds M and N , we will use solutions of the one-dimensional heat equation on M which we thus must understand first. For this we will define and construct a heat kernel or fundamental solution and show the required properties. The original work was done by Minakshisundaram and Pleijel in [8] and completed by Minakshisundaram in [7]. A good reference for this material is Isaac Chavel’s book [2] and an approach that is geared towards spectral geometry may be found in [1]. Also Rosenberg’s book [9] contains a readable introduction.

5.1 Definition Of The Heat Kernel And Its Properties Before we actually construct a heat kernel, we will define a heat kernel by a few conditions, and then show that these already define a unique heat kernel. Afterwards, we will construct a heat kernel satisfying these conditions and thus being the only heat kernel. In this we follow Chavel’s book [2] closely. The heat equation on the Riemannian manifold (M, g) is

0 = ∂tu − ∆g u = Lu, u : M × (0, ∞) 7→ R, (5.1) where u is jointly continuous, twice differentiable in space (first coordinate) and once differentiable in time (second coordinate), and ∆g is the Laplace- Baltrami operator, and L = ∂t − ∆g is a short hand for the heat operator. In this we have not specified boundary conditions, which we need for a specific problem, i.e. we would impose that u tends to the initial data in an appropriate sense as t ↓ 0. Now define the fundamental solution or heat kernel H.

Deﬁnition (Fundamental solution of the heat equation). Let (M, g) be a Riemannian manifold. Then H : M × M × (0, ∞) 7→ R is a fundamental solution of the heat equation (5.1) if

1. H is jointly continuous, C2 in both space variables, and C1 in time,

2. for every Q ∈ M, the restriction H(·, Q, ·) solves the heat equation, i.e. ∂tH − ∆g H = 0,

14 3. for every bounded continuous function f : M 7→ R and Q ∈ M Z lim H(P, Q, t)f(P )dVg(P ) = f(Q). t↓0 M

Note that we impose the heat equation and the convolution with respect R to the ﬁrst variable, i.e. even though H(P, Q, t)f(Q)dVg(Q) for a suitable function f satisﬁes the heat equation, we do not know yet that it tends to f as t ↓ 0. In order to overcome this and to construct solutions for the heat equations from this, we aim to show that the heat kernel is symmetric and is unique. For this we will use Duhamel’s principle.

Proposition 5.1 (Duhamel’s principle). Let u, v ∈ C0(M × (0, t)), and C2 in space, and C1 in time. Then for all [a, b] ⊂ (0, t) Z [ut−b(P )vb(P ) − ut−a(P )va(P )]dVg(P ) M Z b Z = [ut−s(P )(Lvs)(P ) − (Lut−s)(P )vs(P )] dVg(P )ds a M Proof. By expansion of the heat operator ﬁnd for s ∈ [a, b]

ut−s(P )(Lvs)(P ) − (Lut−s)(P )vs(P )

= −ut−s(P ) ∆g vs(P ) + ∆g ut−s(P )vs(P ) + ∂s [ut−s(P )vs(P )] .

2 Since ut−s(·) and vs(·) are C , find over a compact manifold without boundary (Green’s theorem) Z Z ut−s(P ) ∆g vs(P )dVg(P ) = − ∇ut−s(P )∇vs(P )dVg(P ). M M R R b Hence integrating the expansion beforehand over M dVg(P ) and a ds gives the result. With this we can prove the symmetry and uniqueness of the heat kernel which also justifies the previous definition.

Lemma 5.2. A fundamental solution H is symmetric in space, i.e. H(P, Q, t) = H(Q, P, t), and is unique.

15 0 0 Furthermore, if for f ∈ Cb (M) and F ∈ Cb (M × (0, ∞) a solution u ∈ C0(M × (0, ∞)) with C2 in space and C1 in time of ( Lu = F,

limt↓0 ut(·) = f(·) uniformly, exists, it is given by

Z Z t Z ut(P ) = H(P, Q, t)f(Q)dVg(Q) + H(P, Q, t − s)Fs(Q)dVg(Q). M 0 M

Proof. For two fundamental solutions H1 and H2 ﬁx Q, R ∈ M and t ∈ (0, ∞). Plugging u :(P, t) 7→ H1(P, Q, t) and v :(P, t) 7→ H2(P, R, t) into Duhamel’s principle shows for [α, β] ⊂ (0, t) Z Z H1(P, Q, t−β)H2(P, R, β)dVg(P ) = H1(P, Q, t−α)H2(P, R, α)dVg(P ). M M In this we want to take α ↓ 0 and β ↑ t. Since M is compact and H2 is jointly continuous, there exists for every > 0, some β¯ < t such that for all β ∈ [β,¯ t)

|H2(P, R, β) − H2(P, R, t)| < .

Hence Z

lim sup H1(P, Q, t − β)H2(P, R, β)dVg(P ) − H2(Q, R, t) β↑t M Z

≤ lim sup H1(P, Q, t − β)H2(P, R, t)dVg(P ) − H2(Q, R, t) β↑t M Z

+ lim sup H1(P, Q, t − β)dVg(P ) β↑t M = 0 + where we used the point-wise limit assumption of a heat kernel in the last step. Since was arbitrary, this shows Z lim H1(P, Q, t − β)H2(P, R, β)dVg(P ) = H2(Q, R, t) β↑t M

16 and likewise Z lim H1(P, Q, t − α)H2(P, R, α)dVg(P ) = H1(R, Q, t). α↓0 M

Therefore, H1(R, Q, t) = H2(Q, R, t). By letting H1 and H2 the same heat kernel, this shows that every heat kernel must be symmetric. Then the equality for diﬀerent heat kernel shows uniqueness. Finally, for a solution u of the initial value problem, take v :(Q, t) 7→ H(P, Q, t) and plug it into Duhamel’s principle as before. By the same limit argument for α and β the claimed result follows. The heat kernel deﬁnes an evolution operator whose properties we collect in the following theorem.

Proposition 5.3. Suppose the compact manifold (M, d) has a heat kernel 2 H. Deﬁne the operator family It on L (M) for t > 0 by Z It(f)(·) = H(·, Q, t)f(Q)dVg(Q). M

For a ﬁxed f ∈ L2(M), the function u : M × (0, ∞) 7→ R with

u(P, t) = It(f)(P ) satisfies the heat equation (5.1) and for fixed t > 0 the function It(f) has as much regularity as H(·, ·, t). The operator family satisfies the semi-group property, i.e. for t, s > 0 It ◦ Is = It+s.

For each t > 0, the operator It is positive, self-adjoint, and compact. For a ﬁxed f ∈ L2(M) the following limit holds in L2(M)

Itf → f as t ↓ 0.

If f is continuous, the limit holds uniformly.

Proof. By the assumed regularity of H and the compactness of M, we can differentiate twice in space and once in time by differentiating under the integral sign showing that u satisfies the heat equation.

17 2 For a ﬁxed s > 0 and f ∈ L (M), the function Jt+s(f) satisﬁes the heat equation with uniform limit as t ↓ 0, so that by the previous lemma it can by expressed as convolution with the heat kernel, i.e.

It+s(f) = It(Is(f)) showing the semi-group property. Since H is symmetric, the operator is self-adjoint and by continuity of 2 H the image of the unit ball in L (M) under It is equicontinuous and thus precompact by Arzela-Ascoli. By the semi-group property for each f ∈ L2(M)

2 hf, Ptfi = hPt/2f, Pt/2fi = kPt/2fk showing that Pt is positive, so that H must be non-negative everywhere. Since the integral over the divergence is zero, the heat equation shows that for all P ∈ M Z H(P, Q, t)dVg(Q) M is constant in time. By the point-wise limit as t ↓ 0 it must be 1. For any P ∈ M, we can deﬁne a continuous function f by f(Q) = min(1, d(P,Q)/δ) for small enough δ. The point-wise limit shows together with the non- negativity of H that for all > 0, there exists some T > 0 so that for all t ∈ (0,T ) Z H(P, Q, t)1d(P,Q)>δdVg(Q) < . M By compactness of M we can ﬁnd for any small enough δ and > 0, some T such that the for all P,Q ∈ M Z H(P, Q, t)1d(P,Q)>δdVg(Q) < . M Since any continuous f on M is uniformly continuous this then shows the uniform convergence for continuous functions. The limit in L2(M) follows now from the density of continuous functions in L2(M) together with the previous noted fact that Z H(P, Q, t)dVg(Q) = 1. M

18 From the positivity and the integral in the second variable we can formulate the following corollary.

1 Corollary 5.4. Restricting the functions to C (M), the operator It maps C1(M) to C1(M) with kItfkC1 ≤ kfkC1 Proof. Since f is diﬀerentiable, we can integrate by parts using the symmetry of H to control Z

|∇Itf(P )| = H(P, Q, t)∇f(Q)dVg(Q) ≤ sup |∇f(P )| M P ∈M R since H is positive and M H(P, Q, t)dVg(Q) = 1. Likewise the supremum cannot increase.

5.2 Construction of the Heat Kernel In the following we will construct the heat kernel H and show that it is smooth, i.e. H ∈ C∞(M × M × (0, ∞)). In this we follow the books [9, 2]. In order to construct the heat kernel H, we construct an approximate heat kernel S which behaves suﬃciently well as t ↓ 0. In this case we expect the heat kernel to be concentrated around the diagonal so that we try a local approximation. Once we have found such an approximate solution, we consider the error

K = LS.

Hence for some initial data f ∈ L2(M) we expect formally

L(S ∗ f) = (K ∗ f) where ∗ is the convolution as Z (A ∗ f)(P, t) = A(P, Q, t)f(Q)dVg(Q). M Thus we expect with the heat kernel H

S ∗ f = (H + H ∗ K) ∗ f

19 with Z t Z (H ∗ K)(P, Q, t) = H(P, R, t − s)K(R, Q, s)dVg(R)ds 0 M so that formally S H = . 1 + K If the initial approximation S is good enough, then K should be small, i.e. we can deﬁne H as series by

∞ X H = S + S ∗ Q with Q = (−1)nK∗n n=1 and where K∗1 = K and K∗(n+1) = K ∗ K∗n. We now start constructing the approximate solution S, whose properties we control along the diagonal M × M. In order to perform this construction we use spherical exponential coordinates. For this recall the exponential map exp : T M 7→ M given by

expQ X = γ(1) where γ : R 7→ M is a geodesic with γ(0) = Q and γ0(0) = X where X ∈ TQM. By considering the ODE solved by the geodesics, it follows that exp is a smooth map.

For a neighbourhood BTQM () with > 0 the exponential map defines a diffeomorphism. By compactness of M, we can find > 0 such that for every point Q ∈ M the exponential map expQ restricted to BTQM () is a diffeomorphism. Let U ⊂ M × M be the neighbourhood of the diagonal in M × M defined by

(P,Q) ∈ U if P ∈ expQ(BTQM ()).

By the deﬁnition of the distance all pairs (P,Q) 6∈ U are at least apart. For any (P,Q) ∈ U we can construct spherical exponential coordinates m around Q by using spherical coordinates (r, θ1, . . . , θm−1) on R ' TQM and using the exponential map.

Then ﬁnd (r, θ1, . . . , θm−1) such that expQ(r, θ1, . . . , θm−1) = P . By the construction of the exponential map the distance d(P,Q) equals r and by

20 Gauss lemma h∂r, ∂θi i = 0 for i = 1, . . . , m − 1. For the associated volume √ m element gS, we ﬁnd by comparison with the spherical coordinates in R that √ g φ(P,Q) = S rm−1 ∞ deﬁnes a function φ ∈ C (U). ¯ ∞ With this setup we can formulate a local approximation Sk ∈ C (U × (0, ∞)) of order k by

¯ k Sk(P, Q, t) = G(P, Q, t)(u0(P,Q) + u1(P,Q)t + ··· + uk(P,Q)t )

∞ m where G ∈ C (U × (0, ∞)) is the adaption of the heat kernel in R , i.e. d2(P,Q) G(P, Q, t) = (4πt)−m/2 exp − , 4t

∞ and u0, . . . , uk ∈ C (U) are corrections. We now ﬁx the number of corrections k and will show that for large enough k our procedure indeed constructs the heat kernel H. In order to determine the correction, understand the action of the heat operator L in spherical harmonic coordinates at Q ∈ M. For this recall that r = d(P,Q) so that

m r2 ∂ S¯ = − + G (u + ··· + u tk) + G (u + ··· + ku tk−1) t 2t 4t2 0 k 1 k and acting on the ﬁrst variable P

¯ k k k ∆ Sk = (∆ G)(u0+···+ukt )+2h∇G, ∇(u0+···+ukt )i+G ∆(u0+···+ukt )

0 where with φ = ∂rφ

√ 2 0 ∂r( gS∂rG) r m r φ ∇G = √ = 2 − − G gs 4t 2t 2t φ and using Gauss lemma r h∇G, ∇(u + ··· + u )i = − ∂ u + ··· + ∂ u tk G. 0 k 2t r 0 r k

21 Therefore, we ﬁnd h r φ0 LS¯ = G (u + ··· + ku tk−1) + (u + ··· + u tk) k 1 k 2t φ 0 k r i + (∂ u + ··· + ∂ u tk) − (∆ u + ··· + ∆ u tk) . t r 0 r k 0 k

Hence we want to choose u0, . . . , uk such that the bracket vanishes except for the highest order of t, i.e. the correction should satisfy r φ0 u + r∂ u = 0 2 φ 0 r 0 and for l = 1, . . . , k r φ0 l + u + r∂ u = ∆ u . 2 φ l r l l−1

We can ensure the existence of smooth functions u0, . . . , uk satisfying these conditions by solving the ODEs from r = 0. In this we also impose that u0(Q, Q) = 1 for all Q ∈ M in order to match the heat operator which we will need in showing the limit for the convolution. With this we explicitly ﬁnd recursively 1 u (P,Q) = √ , 0 φ Z 1 1 lp ul(P,Q) = √ τ φ ∆ ul−1dτ, φ 0 where the integrand is evaluated at (rτ, θ1, . . . , θm). ∞ As φ is a positive smooth function indeed u0, . . . , uk ∈ C (U) and ¯ k LSk = G ∆ ukt . (5.2)

¯ ∞ ∞ Now extend Sk ∈ C (U × (0, ∞)) to Sk ∈ C (M × M × (0, ∞)) by using a smooth cutoﬀ η ∈ C∞(M × M) only depending on the distance r = d(P,Q) with η(r) ≡ 1 for r ≤ /2 and η(r) ≡ 0 for r ≥ . Then Sk is indeed an approximate heat kernel for k > m/2 which we formalise as parametrix. Deﬁnition. A function S ∈ C∞(M × M × (0, ∞)) is a parametrix for the heat equation if

22 1. LS ∈ C0(M × M × [0, ∞)), i.e. continuous up to t = 0,

2. for all f ∈ Cb(M) Z lim S(P, Q, t)f(P )dVg(P ) = f(Q). t↓0 M

Lemma 5.5. If k > m/2, then Sk is a parametrix. If k > (m/2) + 2l, then l LSk ∈ C (M × M × [0, ∞)).

2 Proof. Outside U/2 the cutoff is smooth and the exponential exp(−r /(4t)) in G ensures that it remains smooth up to t = 0. Inside U/2 we find k LSl = G ∆ ukt which converges to 0 as t & 0 if k > m/2. Likewise the lth derivative exists and is bounded if k > (m/2) + 2l. (Note that for the spatial derivatives we could just impose k > (m/2) + l.) For the limit of convolution use the exponential coordinates φQ around Q to find Z lim Sk(P, Q, t)f(P )dVg(P ) t↓0 M Z 2 −m/2 −r k √ = lim (4πt) exp (u0 + ··· + ukt )f(φQ(x)) gφQ dx t↓0 m 4t B(R )

= u0(Q, Q)f(Q) = f(Q) by using the properties of the standard heat kernel on Rm. For constructing the series we need some ﬁner control on the error

Kk = LSk.

By compactness of M, the control of the exponential, and (5.2) we can ﬁnd for any ﬁxed time T a constant A such that for t ∈ [0,T ] and P,Q ∈ M

k−m/2 |K(P, Q, t)| ≤ At ≤ B with B = AT k−m/2.

23 By induction we can then estimate

ABn−1V n−1tk−(m/2)+n−1 |K∗n(P, Q, t)| ≤ k (k − (m/2) + 1)(k − (m/2) + 2) ... (k − (m/2) + n − 1) (5.3) where V is the volume of M. For n = 1 this is just the above estimate. Assuming it is true for n, ﬁnd

∗(n+1) |Kk (P, Q, t)| Z t Z ∗n ≤ |K(P, R, t − s)K (R, Q, s)|dVg(R)ds 0 M Z t Z ABn−1V n−1sk−(m/2)+n−1 ≤ B · ds 0 M (k − (m/2) + 1)(k − (m/2) + 2) ... (k − (m/2) + n − 1) ABnV n Z t = sk−(m/2)+n−1ds (k − (m/2) + 1)(k − (m/2) + 2) ... (k − (m/2) + n − 1) 0 ABnV ntk−(m/2)+n = . (k − (m/2) + 1)(k − (m/2) + 2) ... (k − (m/2) + n)

This proves the estimate for n + 1 and thus for all n by induction. P∞ ∗n Hence when k > m/2, the series n=1 |Kk | converges and thus Ql = P∞ n ∗n 0 n=1(−1) Kk converges to a function in C (M × M × [0, ∞)) which converges uniformly to 0 as t ↓ 0. l By the same estimate as in lemma 5.5, indeed Qk ∈ C (M × M × [0, ∞)) for k > (m/2) + 2l and the diﬀerentiation can be done term by term. In order to verify that Hk = Sk + Sk ∗ QK satisﬁes the heat equation for large enough k we use the following lemma.

l l Lemma 5.6. Let F ∈ C (M×M×[0, ∞)), then Sk ∗F ∈ C (M×M(0, ∞)). If l ≥ 2, then L(Sk ∗ F ) = F + Kk ∗ F. and there exists a constant C such that for all P,Q ∈ M and t ∈ [0, 1]

|∇(Sk ∗ F )(P, Q, t)| < C where ∇ is acting on the ﬁrst coordinate.

Proof. In a neighbourhood of P ∈ M identify the tangent spaces TP¯M in m a smooth way with R and let φP¯ : B() 7→ M be expP¯, i.e. exponential

24 coordinates. The smooth identiﬁcation can e.g. be done by parallel transport on geodesics stating at P . Then with the cutoﬀ η we can express the convolution as Z t Z ¯ −m/2 −|x|2/4s (Sk ∗ F )(P , Q, t) = (4πs) e 0 B() ¯ ¯ k (u0(P , φP¯(x)) + ··· + uk(P , φP¯(x)s )η(|x|)

F (φP¯(x), Q, t − s)dxds. Since Z t Z (4πs)−m/2e−|x|2/4sdxds < ∞ 0 B() l we can diﬀerentiate under the integral sign and ﬁnd Sk ∗ F ∈ C (M × M × (0, ∞)). By the compactness of M × M × [0, 1], this also shows that there exists a constant C such that

|∇(Sk ∗ F )(P, Q, t)| < C for all P,Q ∈ M and t ∈ [0, 1]. In order to show the action of the heat operator, we split the time-integral and use integration by parts, as we would do for the standard heat kernel in Rm (cf. [4, Section 2.3]). Hence for a small δ > 0 Z L(Sk ∗ F )(P, Q, t) = Sk(P, R, δ)F (R, Q, t − δ)dV (R) M Z t Z + (LSk)(P, r, s)F (R, Q, t − s)dV (R)ds δ M + Iδ where Z δ Z −m/2 −|x|2/4s Iδ(P, Q, t) = (4πs) e 0 B() k L (u0(P, φP (x)) + ··· + uk(P, φP (x)s )η(|x|)F (φP (x), Q, t − s) dxds.

By the assumed regularity, Iδ → 0 as δ → 0. Hence by the limit δ → 0, we ﬁnd the claimed relation

L(Sk ∗ F ) = F + Kk ∗ F.

25 Therefore, Hk = Sk + Sk ∗ Qk for k > (m/2) + 4 satisﬁes indeed the heat equation as

∞ ∞ X n ∗n X n ∗n LHk = Kk + Kk ∗ (−1) Kk + (−1) Kt = 0. n=1 n=1

Since Qk also tends to 0 uniformly as t ↓ 0 for k > m/2, we still have for any f ∈ Cb(M) Z lim H(P, Q, t)f(P )dVg(P ) = f(Q). t↓0 M

This shows that for Hk is a heat kernel for k > (m/2) + 4 and is in Cl(M × M × (0, ∞)) for k > (m/2) + 2l. By the uniqueness of the heat kernel all Hk for k > (m/2) + 4 agree and thus define the heat kernel H with infinite regularity. Hence we have shown the first half of our main theorem.

Theorem 5.7. Given a compact Riemannian manifold (M, g), there exists a fundamental solution H ∈ C∞(M × M × (0, ∞)) of the heat equation. Furthermore, there exists a constant C such that for P ∈ M and t ∈ [0, 1]

Z t Z |H(P, Q, s)|dVg(Q)ds ≤ Ct 0 M and Z t Z √ |∇H(P, Q, s)|dVg(Q)ds ≤ C t 0 M where ∇ is acting on the ﬁrst variable.

Proof. It only remains to show the integral bound. By the previous lemma, ˜ ﬁnd a constant C1 such that Z t Z ˜ |(Sk ∗ Qk)(P, Q, s)|dVg(Q)ds ≤ C1V t 0 M and Z t Z √ ˜ |∇(Sk ∗ Qk)(P, Q, s)|dVg(Q)ds ≤ C1V t 0 M where V is the volume of M.

26 k ˜ Since η(d(P,Q))(u0 + ··· + ukt ) is bounded, ﬁnd for some constant C2

Z t Z Z t Z ˜ −m/2 −|x|2/4t ˜ |Sk(P, Q, s)|dVg(Q)ds ≤ C2 (4πt) e dxdt = C2t. 0 M 0 Rm

k Also ∇[η(d(P,Q))(u0 + ··· + ukt )] is bounded, so

Z t Z Z t Z ˜ −m/2 −|x|2/4t |∇Sk(P, Q, s)|dVg(Q)ds ≤ C3 (4πt) e dxdt 0 M 0 Rm Z t Z ˜ −m/2 |x| −|x|2/4t + C4 (4πt) e dxdt 0 Rm 2t ˜ ˜ for some constants C3 and C4. The ﬁrst integral is again t and for the second integral we ﬁnd by using spherical coordinates Z −m/2 |x| −|x|2/4t ˜ −1/2 (4πt) e dx ≤ C5t Rm 2t ˜ for some constant C5. Putting, these bounds together shows the claimed control.

27 6 Short time existence

Our approach to long and short time existence follows the book of Lin & Wang, [6], but is adapted to work for the domain being an arbitrary Rie- mannian manifold M which is compact and without boundary, rather than the case M = Rn. To prove that (4.13) must always admit a local solution, we will translate the problem to that of finding a fixed point of an integral operator. When restricted to a set of functions which stay close to the initial data u0, this integral operator is a strict contraction. An application of Banach’s Fixed Point Theorem then yields the existence of a unique local solution. This method also yields a blow up criterion for the non existence of global solutions, which is achieved by showing that the length of time for which a local solution exists is bounded below by a function of the norm of the initial data. Given a heat kernel H(P, Q, t) ∈ C(M×M×(0, ∞)) for the heat equation L L on M, we can construct an evolution operator It : C(M, R ) 7→ C(M, R ), defined by Z It(u)(P ) = H(P, Q, t)u(Q) dVg(Q), M L which satisfies the heat equation for any initial data u0 ∈ C(M, R ). Via Duhamel’s principle, we can use It to convert the equation (4.13) into an integral equation as follows: Multiplying (4.13) by the integrating factor It−s, we obtain d (I u ) = I A(u )(∇u , ∇u ). ds t−s s t−s s s s Integrating with respect to s from 0 to t then leaves us with Z t ut = Itu0 + It−sA(us)(∇us, ∇us) ds (6.1) 0 as required. Hence, to find a solution of (4.13), it suffices to find a fixed point of the map Z t T : v 7→ Itu0 + It−sA(vs)(∇vs, ∇vs) ds. 0

We wish to study the action of T on functions v which are in C1(M, RL) for each t ∈ [0, ) (where is to be specified later) so that the term A(vt)(∇vt, ∇vt) is pointwise well defined and continuous for fixed t. However, our initial data

28 L u0 is only assumed to be in C(M, R ) and so the diﬀerentiability of T v at t = 0 is not guaranteed. This issue is easily remedied by the following lemma, ∞ L which shows that we can always assume that u0 ∈ C (M; R ). Lemma 6.1. Every v ∈ C(M, N ) is homotopic to a function v˜ ∈ C∞(M, N ).

Proof. By compactness of M, there exists some > 0 such that, for t ≤ , (Itv)(M) is contained within the domain of deﬁnition of π, the nearest point projection operator from the tubular neighbourhood Nδ0 to N . Hence, the map t 7→ (π ◦ It)v describes a homotopy within C(M, N ) between v and ∞ v˜ := (π ◦ Iε)v. The fact thatv ˜ ∈ C (M, N ) now follows from the fact that π is a smooth map combined with Proposition 5.3. Deﬁne ∞ 1 L X := L ([0, ),C (M, R )). (6.2) It is clear that, when endowed with the norm

kuk = sup ku k 1 L , X t C (M,R ) t∈[0,)

X is a Banach Space. Theorem 6.2. The integral equation (6.1) admits a unique local solution [ u ∈ X,

0≤

Since A is smooth and It is as well for t 6= 0, we have that T (X) ⊂ X and so T is well deﬁned. Now, for δ > 0, deﬁne

B := {v ∈ X : kv − I u k ≤ δ}. (6.4) δ t 0 X

29 Lemma 6.3. For suﬃciently small , δ, T (Bδ) ⊂ Bδ.

Proof. First, we enforce , δ ≤ 1 and that δ ≤ δ0. By expression in local coordinates,

a ij a b c A (v)(∇v)(∇v) = −g π ;bcv ,iv ,j,

and so, by the compactness of Nδ0 and M, it follows that |A(v)(∇v, ∇v)| ≤ 2 C|∇v| for v ∈ Bδ. By Theorem 5.7, we therefore have that

Z t Z

|T v − Itu0|(P ) = H(P, Q, t − s)A(vs)(∇vs, ∇vs)(Q)dVg(Q)ds 0 M Z t Z ≤ sup kA(vs)(∇vs, ∇vs)kC(M) |H(P, Q, t − s)|dVg(Q) s∈[0,t] 0 M ≤ C sup k∇v k2 ≤ C kvk2 , s C(M) X s∈[0,t] and similarly that

Z t Z

|∇(T v − Itu0)|(P ) = ∇H(P, Q, t − s)A(vs)(∇vs, ∇vs)(Q)dVg(Q)ds

√0 M ≤ C kvk2 . X

It follows that √ kT v − I u k ≤ C kvk2 . t 0 X X

Note that, for v ∈ Bδ,

kvk ≤ kI u k + kv − I u k ≤ 1 + kI u k , X t 0 X t 0 X t 0 X1 and so, upon taking √ δ = C 1 + kI u k 2 t 0 X1 we are able to guarantee that T (Bδ) ⊂ (Bδ).

Lemma 6.4. For small enough , the map T is a strict contraction on Bδ:

kT u − T vk ≤ λ ku − vk for all u, v ∈ B , X X δ for some λ ∈ (0, 1).

30 Proof. First, we will show that the map v 7→ A(v)(∇v, ∇v) is locally Lips- chitz from C1(M, RL) to C(M, RL). Recalling that, in local coordinates on a ij a b c M, A (v)(∇v, ∇v) = −g π ,bcv ,iv ,j we have that

a a ij a b c a b c A (u)(∇u, ∇u) − A (v)(∇v, ∇v) = − g (π ,bc(u)u ,iu ,j + π ,bc(v)v ,iv ,j) ij a b c b c = − g (π ,bc(u)(u ,iu ,j − v ,iv ,j) a a b c − (π ,bc(u) − π ,bc(v))v ,iv ,j).

To control the ﬁrst term, we use the fact that gij = gji and that (since π is a a smooth) π ,bc = π ,cb to obtain

ij a b b c c a ij b c b c g π ,bc(u)(u ,i + v ,i)(u ,j − v ,j) = π ,bc(u)(g (u ,iu ,j − v ,iv ,j) ij b c b c + g (v ,iu ,j − u ,iv ,j)) ij a b c b c = g π ,bc(u)(u ,iu ,j − v ,iv ,j) ij a b c ji a c b + g π ,bc(u)v ,iu ,j − g π ,cb(u)v ,ju ,i ij a b c b c = g π ,bc(u)(u ,iu ,j − v ,iv ,j).

a L For the second term, since π ,bc is a diﬀerentiable map from R to R we can a apply the mean value theorem to the function π ,bc(tu + (1 − t)v) to obtain

a a a d d |π ,bc(u) − π ,bc(v)| ≤ sup (|π ,bcd|(x)) |u − v |. (6.5) x∈Nδ0

a N is compact, so Nδ0 is as well, so |π | and all of its derivatives are bounded by a constant depending only on Nδ0 and we therefore have

|Aa(u)(∇u, ∇u) − Aa(v)(∇v, ∇v)| ≤ C (1 + |∇u| + |∇v|) |∇u − ∇v| , which is exactly the statement that v 7→ A(v)(∇v, ∇v) is locally Lipschitz. It therefore follows from Theorem 5.7 that Z t |T ut − T vt| ≤ |It−s (A(us)(∇us, ∇us) − A(vs)(∇vs, ∇vs)| ds 0 Z t ≤ C sup kA(vs)(∇vs, ∇vs) − A(us)(∇us, ∇us)kC(M) |It−s| ds s∈[0,t] 0 ≤ C sup 1 + k∇uskC(M) + k∇vskC(M) k∇us − ∇vskC(M) , s∈[0,t]

31 and similarly that √ |∇T ut−∇T vt| ≤ C sup 1 + k∇uskC(M) + k∇vskC(M) k∇us − ∇vskC(M) . s∈[0,t]

Taking the supremum in P and t and combining these two inequalities leaves us with √ kT v − T uk ≤ C 1 + kuk + kvk ku − vk X √ X X X ≤ C 1 + kI u k ku − vk , t 0 X1 X Hence, by taking ( ) √ δ 1 = min , , 2 C 1 + kI u k C 1 + kItu0k t 0 X1 X1 we can guarantee that T is a strict contraction from Bδ to Bδ with contraction 1 constant λ ≤ 2 . By Banach’s Fixed Point Theorem, it follows that T admits a fixed point u ∈ Bδ and so, following on from the discussion in the introduction to this section, that (4.13) is guaranteed a short time solution valid for a time period of length at least . From Proposition 5.3 and Equation (6.1) we can then ∞ L deduce that ut ∈ C (M × (0, ∞), R ) and from 4.2 we have that ut ∈ C∞(M × (0, ∞), N ). Now, assume that ut is a solution to (4.13) defined on a finite maximal interval of existence [0,Tmax) and that kutk < ∞. Let t0 ∈ [0,Tmax) and XTmax consider the new heat flow ( ∂ v − ∆ v = A(v )(∇v , ∇v ) t t g t t t t (6.6) v0 = ut0 .

By the discussion following Lemma 6.4, the equation (6.6) admits a unique local solution, valid for a time period of length at least , where   √  δ 1  = min , . 2  C 1 + kI u k  C 1 + kItut0 k t t0 Xt  Xt0+1 0+1

32 By Corollary 5.4, we have that kItut0 k ≤ kut0 k 1 L . This implies Xt0+1 C (M,R ) that is bounded uniformly away from 0 for t0 ∈ [0,Tmax), by 0, say. Taking t0 to be such that Tmax−t0 < 0, then, we see that ut can be continued in a way which satisﬁes equation 4.13 beyond t = Tmax by deﬁning ut0+s = vs. This contradicts the maximality of the interval [0,Tmax) and hence demonstrates that, if T < ∞, we must have that ku k → ∞ as t ↑ T . Since max t C1(M,RL) max u takes values within the compact manifold w(N ), ku k is uniformly t t C(M,RL) bounded. Hence, we must in fact have that k∇u k → ∞ as t ↑ T . t C(M,RL) max

7 Global solution to the heat ﬂow

In this section we will aim to prove the main result of Eells and Sampson’s seminal paper [3].

Theorem 7.1. If M is compact and N has non-positive sectional curvature then the heat ﬂow ( ∂ f = τ(f ), t t t (7.1) f(·, 0) = f0,

∞ admits a solution ft ∈ C (M, N ) valid for all time. Further we can find a sequence tk ↑ ∞ such that ftk → f∞ uniformly in C(M, N ) as k → ∞, where f∞ is harmonic. In order to to prove this theorem will will first derive a number of estimates which we will use to show that we have a solution to (7.1) for all time, and further so that we may apply a suitable compactness result to show that we have convergence and also to show that the limiting function f∞ is in fact a harmonic map. Rather that the entire heat flow (ft) converges, however, we will only show that a subsequence of samples (ftk ) converges as tk ↑ ∞. We will then use the compactness of M to argue that eventually ftk is so close to f∞ that each point ftk (P ) is connected to f∞(P ) by a unique geodesic and hence that ftk and f∞ are homotopic via a geodesic flow.

7.1 A priori estimates Here we derive several results that will be fundamental to the proof of theorem 7.8. Recall, that the tension ﬁeld τ(f) is the gradient of E(f) in the sense

33 that R dft Mhτ(f), dt t=0idVg gives the directional derivative of E(·) at f in the di- dft rection dt t=0. If ft satisﬁes the heat ﬂow ∂tft = τ(ft), we will have d Z (E(ft)) = − hτ(ft), ∂tftidVg dt M Z 2 = − |∂tft| dVg M Z 2 = − |τ(ft)| dVg ≤ 0 (7.2) M and so E(ft) will decrease as t → ∞. Furthermore E(ft) is strictly decreasing unless τ(ft) = 0 (whence ft is harmonic). Since the energy is a quadratic form, E(ft) is bounded below. We will, however, have to work a little harder to show that limt→∞ E(ft) = inft∈[0,∞) E(ft).

Lemma 7.2 (Energy Inequality). For any 0 < T ≤ ∞, if ft solves the heat equation on M × [0,T ], then

Z t 2 0 ≤ E(ft) = E(f0) − kτ(fs)kL2(M)ds ≤ E(f0), 0 < t < T. (7.3) 0 Proof. Integrating (7.2) yields the desired inequality. We prove a ﬁrst estimate.

3,1 Proposition 7.3 (Bochner Identity). If ft ∈ C (M × [0,T ), N ) satisﬁes the heat ﬂow then

2 M ij α β N α γ β δ ij kl (∂t − ∆g)e(ft) + |∇df| + (Ric ) ft,ift,jhαβ = (R )αβγδft,ift,jft,kft,lg g where ∇ is the covariant derivative on T ∗M ⊗ f −1T N , RicM is the Ricci tensor of M and RN is the Riemann tensor of N .

Taking a stationary ft reduces to the Bochner inequality for harmonic maps. Proof. We prove this pointwise for every P ∈ M and t ∈ [0,T ). For this we choose local normal coordinates on M and N centered at P and ft(P ).

34 In these we ﬁnd 1 (∆ −∂ )e(f ) = (∆ −∂ )gijh f α f β g t t 2 g t αβ t ,i t ,j 1 = gijgkl( h f γ f δ f α f β + h f α f β ) 2 αβ,γδ t ,k t ,l t ,i t ,j αβ t ;ik t ;jl α β α β + hαβft ,ift ;jkl − hαβft ,i(∂tft ),j). By the Ricci identity

β β M pq β ft ;jkl = ft ;klj + (R )lpkjg ft ,q.

As ft satisﬁes the heat equation we ﬁnd

ij kl α β α β g g (hαβft ,ift ;jkl − hαβft ,i(∂tft ),j) ij kl α N β γ δ = g g hαβft ,i(−(Γ )γδft ,kft ,l),j ij kl N β α γ δ = −g g hαβ(Γ )γδ,ft ,ift ,kft ,lft ,j. Hence we arrive at

2 M ij α β (∂t − ∆g)e(ft) + |∇dft| + (R ) ft ,ift ,jhαβ 1 = gijgklf α f β f γ f δ ((ΓN ) h − h ). (7.4) t ,i t ,j t ,k t ,l γδ,β α 2 αβ,γδ Recall the Christoﬀel symbols 1 (ΓN )α = hαδ(h + h − h ). βγ 2 βδ,γ γδ,β βγ,δ

In normal coordinates we ﬁnd at ft(P ) for the Riemmann tensor

N α N α N α (R ) βγδ = (Γ )βδ,γ − (Γ )βγ,δ 1 = hα(h + h − h − h ). 2 δ,βγ βγ,δ βδ,γ γ,βδ On the other hand for our bracket in (7.4) 1 1 (ΓN ) h − h = (h + h − h − h ) = R γδ,β α 2 αβ,γδ 2 γα,δβ δα,γβ αβ,γδ γδ,αβ αγβδ which proves our result.

35 Corollary 7.4. Suppose that M is compact and that N has non-positive N sectional curvature, that is, K ≤ 0. If ft solves the heat equation then we have (∂t − ∆g)e(ft) ≤ Ce(ft), (7.5) for some C > 0, which depends only on the Ricci curvature of M.

Proof. Using that RN ≤ 0 this follows from the previous proposition 7.3. We now state the following useful result from [5].

Proposition 7.5 (Moser’s Harnack Inequality for Subsolutions of the Heat ∞ 2 Equation). Let v ∈ C (BR(x0) × [t0 − R , t0]) be a non-negative function satisfying ∂v − ∆v ≤ C v, ∂t 1 2 on the cylinder BR(x0) × (t0 − R , t0], where C1 ∈ R. Then there exists a constant C2 such that

Z t0 Z −(m+2) v(x0, t0) ≤ C2R v(y, s)dVgds. 2 t0−R BR(x0) For completeness we include the following result, which we do not use in the proof of Theorem 7.1, however it does show that any stationary point of the energy functional (i.e. harmonic map) that we ﬁnd is energy minimising.

Lemma 7.6 (Convexity of E(ft)). If N has non-positive sectional curvature d2 then dt2 E(ft) ≥ 0.

DN β Proof. Denote by dt the covariant derivative (considering ft,j as a tangent vector in N ):

d DN e(f ) = h f α f β gij dt t αβ t,i dt t,j d2 DN DN DN DN e(f ) = h f α f β gij + h f α f β gij. dt2 t αβ t,i dt dt t,j αβ dt t,i dt t,j

36 As the connection is torsion-free (where ∇N denotes the covariant derivative on N ):

DN f α = ∂ f β∇N f α dt t,i t t β t ,i α ∂ft = ∂ f β ,i + Γα f γ t t ∂yβ γβ t ,i N α = ∇i ∂tft , and using the Ricci identity,

DN DN ∇N ∂ f α = ∇N ∂ f α + (RN )α ∂ f γf δ ∂ f β. dt i t t i dt t t βγδ t t t ,i t t Furthermore,

N α ij α N α γ δ ij ∇i (hαβft,jg ) = hαβ∆gft + hαβ(Γ )γδft ,jft ,ig

= τ(f)β.

N α Hence, integrating by parts we ﬁnd (since (R ) β[γδ] = 0):

2 Z 1 d E(ft) N α N β ij 2 = hαβ ∇i ∂tft ∇j ∂tft g dVg 2 dt M Z N α γ β δ ij − (R )αβγδft ,ift ,j∂tft,i∂tft g dVg. M

2 d E(ft) Therefore, if N has non-positive sectional curvature, we have dt2 ≥ 0

The above results culminate in the following theorem:

Theorem 7.7. Let ft be a solution of the heat ﬂow (7.1), where N has non-negative sectional curvature. We have the following a priori estimate:

k∇ftkC0(M) ≤ C ∀t ∈ (0,T )

Proof. Firstly, by Corollary 7.4 we have

(∂t − ∆g)e(ft) ≤ Ce(ft)

37 So we may apply Proposition 7.5 to (x0, t0) ∈ M×[1,T ), for R < 1, yielding:

Z t0 Z −(m+2) e(ft0 )(x0) ≤ CR e(ft)dVg dt 2 t0−R M Z t0 −(m+2) = CR E(ft)dt 2 t0−R −m ≤ CR E(f0), by the energy inequality, Lemma 7.2. So |∇ft| is uniformly bounded on M×(1,T ). To bound the gradient on M×[0, 1] we write a = e(ft) exp(−Ct), for which we have the expression

∂ta − ∆ga ≤ 0.

So by the maximum principle, we have a(x, t) ≤ max a(x, 0). That is,

Ct C e(ft)(x) ≤ kf0kC0(M)e ≤ e kf0kC0(M), so choosing the maximum of these two bounds yields the required result. We are now in a position to prove the ﬁrst part of Theorem 7.1.

Theorem 7.8. If M is compact and N has non-positive sectional curvature ∞ then the heat ﬂow (7.1) admits a solution ft ∈ C (M, N ) valid for all time. Proof. By Theorem 7.7, we have that the gradient of u is uniformly bounded on M × (0,T ). Since, by Theorem 6.2 we have that the gradient blows up ∞ as t approaches T , we ﬁnd that we must have T = ∞, so ft ∈ C (M, N ). By a bootstrapping argument, that is, applying the analysis in Theorem 7.7 m−1 to e(∇ ft), m = 2, ... we have

m sup k∇ ftkC0(M) ≤ C(m, M, N , u0), ∀m ≥ 1 (7.6) t∈[0,∞)

So we have a solution that exists for all time, which is smooth.

38 7.2 Convergence at inﬁnity to a harmonic map

Theorem 7.9. There exists a sequence (tk) such that tk ↑ ∞ and ftk → some f∞ uniformly in C(M, N ), where f∞ is homotopic to f0. Further, the limit f∞ is harmonic. Proof. By the energy inequality, Lemma 7.2, we have that for any t > 0, Z t Z 2 |∂tft| dVgdt ≤ E(f0) < ∞. 0 M From this, we may deduce that Z t Z Z t Z Z t−2 Z 2 2 2 lim |∂tft| dVgdt = lim |∂tft| dVgdt − |∂tft| dVgdt = 0. t↑∞ t−2 M t↑∞ 0 M 0 M Since

2 N (∂t − ∆g) |∂tft| = − |∂tft| + R (ft)(∇ft, ∂tft, ∇ft, ∂tft) ≤ 0, we may apply Lemma 7.5 to ﬁnd that for any α ∈ (0, 1),

k∂tftkCα(M×[t−1,t]) ≤ C(α) k∂tftkL2(M×[t−2,t]) → 0, (7.7)

α as t → ∞. This implies that, for all t, ft ∈ C (M) with H¨olderconstant uniformly bounded and so by Arzela-Ascola we ﬁnd some sequence tk ↑ ∞ α such that ∂tftk → 0, by (7.7), in C (M). So we have that f∞ := limk→∞ ftk is harmonic. It can be shown by a further application of Arzela-Ascoli using the bounds found in Theorem 7.7 that, passing to a subsequence if necessary, ftk → f∞ uniformly in C(M, N ). By the uniform convergence we can choose k large enough such that d(ftk (P ), f∞(P )) is less than the radius of injectivity for all P ∈ M. Then the linear homotopy along the unique geodesic between ftk (P ) and f∞(P ) shows that these are homotopic.

Since ftk is homotopic to f0 by the heat ﬂow, we can combine these homotopies to conclude that f0 is homotopic to f∞.

8 Conclusion

This report can serve as an introduction to harmonic maps and their heat- ﬂows. The next natural area of study, after the results given by Eells and

39 Sampson in [3], is the result of Struwe, who, in his paper [10], studied the case where the assumption on the curvature of N is dropped. In this case, the main technical difficulty is blow up in finite time of the heat flow, i.e. as t approaches Tmax ,as in theorem 6.2, the gradient of ft blows up, so that the proof of long time existence presented here is no longer valid. Struwe deals with this by showing that this type of singularity can only occur on a finite set and that, once these singularities are accounted for, the heat flow can still be continued. Heat flow methods have found applications in many other areas of geometry, most notably in Richard Hamilton’s programme for the use of Ricci Flow, where a heat operator is applied to the curvature of the manifold, to solve the Thurston Geometrization conjecture which was famously completed by Perelman in 2003.

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