Heat Flow Methods in Geometry andTopology

H. Dietert, K.Moore, P. Rockstroh, G. Shaw

April 1, 2014

Introduction

For a domain Ω ⊂ Rn, the Dirichlet energy functional∫

Ω|∇f |2dx, defined for

functions f ∈ C1(Ω,R), is a central object of interest in analysis and has beenwidely studied. Its minimisers satisfy the Laplace equation ∆ f = 0 and thischaracterisation shows that the minimisers enjoy many special properties,e.g. infinite differentiability. We can generalise the Dirichlet energy func-tional E(f) to functions f ∈ C1(M,N ) between abstract Riemannian man-ifolds (M, g) and (N , h). The stationary points satisfy the Euler-Lagrangeequation and are the generalisation of harmonic maps.

The first systematic treatment of harmonic maps between manifolds ofarbitrary dimension was undertaken by James Eells and J.H. Sampson intheir seminal paper, ‘Harmonic Mappings of Riemannian Manifolds’ [3]. Thebasic question they tackled was if given a function f ∈ C0(M,N ), thereexists a harmonic function which is homotopic. The main idea is to studythe studying a corresponding heat flow on the manifold under which E(f)must decrease, and by analysing the long time behaviour of solutions, tofind a related harmonic map. They proved that, in the case that the targetmanifold N has non-positive sectional curvature, a solution to the heat flowequation exists for all time, and further, as t → ∞, a harmonic map occursin the limit.

This report gives a self-contained discussion of these results and in par-ticular gives a rigorous complete treatment for the existence of the heat flowin a very general setting. Our exposition is mostly a combination of the pre-sentation in Eells and Sampson’s original paper ([3]) and the presentation in

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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 2introduces the Dirichlet integral for maps between manifoldsM and N anddefines harmonic maps as the critical points of this functional. It is alsoshown 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 thesteady state solution to the heat flow associated with the tension field. Thisheat flow is analysed by using the Nash Embedding Theorem to embed thetension field into Euclidean space as a system of equations valid for mapstaking values in RL in Section 4. Next, in Section 5 the theory for the linearheat equation, valid for scalar valued maps defined on M, is developed. InSection 6, the results of Sections 4 and 5 are combined with a fixed pointargument to prove that the heat flow must always admit a local solutionand 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 Nis assumed to have non-positive sectional curvature, no such blow up canoccur and by then using the existence of a global solution for the heat flowto construct the required homotopy.

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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 145.1 Definition Of The Heat Kernel And Its Properties . . . . . . . 145.2 Construction of the Heat Kernel . . . . . . . . . . . . . . . . . 19

6 Short time existence 28

7 Global solution to the heat flow 337.1 A priori estimates . . . . . . . . . . . . . . . . . . . . . . . . . 337.2 Convergence at infinity to a harmonic map . . . . . . . . . . . 39

8 Conclusion 39

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1 Tools from differential geometry

A smooth m-dimensional manifold M looks, at every point P ∈ M, likeRm, and the relation is smooth. That is, for all P ∈ M, there is a smoothdiffeomorphism 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 inM and denote them as x1, x2, . . . , xm and this coordinate system extends tothe neighbourhood of P on which ϕ is defined.

The idea of tangent space to M at P is intuitively easy to imagine. Youhave come across the tangent to a curve; in this case the tangent space isjust the line tangent to the curve at that point. Now picture a sphere. Picka 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 atangent vector field on a manifold, which sends each point of M to a vectorin its tangent space. We may express a tangent vector field X in terms ofthe basis ∂

∂x1, ∂∂x2, . . . , ∂

∂xmas

X = X i ∂

∂xi, (1.1)

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 imageof a small region in the domain with one system in local coordinates.

A metric g introduces length and angles on the manifold by describing ainner product between two tangent vectors, i.e.

g : TM× TM→ R. (1.2)

With respect to a local coordinate system we may write g|p as a matrix, ifwe plug in the basis vectors to the tangent space at a point p, i.e.

gij :=

⟨∂

∂xi,∂

∂xj

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

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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 iscalled the radius of injectivity. At any point P ∈ M we can construct localnormal coordinates xi using an object called the exponential map, which isconstructed so that at P the metric looks as close to the Euclidean metric aspossible. These coordinates are especially useful for calculations and we willmake use of them.

If we have a vector field X and local coordinates xi, we can define thepartial derivative as

X,i :=∂X

∂xi=∂Xj

∂xi∂

∂xj. (1.3)

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 existsthe unique Levi-Civita connection (which we refer to as the connection fromherein), 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 thisconnection and denote

X;j = ∇jX. (1.4)

In a coordinate basis this takes the form

X i;j =

∂X i

∂xj+ ΓijkX

k, (1.5)

where Γijk are the Christoffel symbols which, for the Levi-Civita connectionare

Γijk =1

2gil(gjl,k + gkl,j − gjk,l). (1.6)

In normal coordinates centered at P , the Christoffel symbols vanish andthe covariant derivative and partial derivative agree when all are evaluatedat P .

If we want to integrate over the manifold we need to find a coordinateindependent way. On a Riemannian manifold, this can with local coordinatefunctions and integration behaves as we would expect. Furthermore, thedivergence 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∫

Mf(P )dVg(P ) =

∫V

f(ϕ(x))√gdx

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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 normalLaplace operator. For this we define

∆g f = gijf;ij. (1.7)

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

∆g f =m∑i=1

∂i(√g∂if)√g

2 Harmonic maps and the energy functional

Consider a smooth map f : (M, g) 7→ (N , h) between smooth Riemannianmanifolds (M, g) and (N , h) where both manifolds are without boundaryand compact. Throughout the report we will assume compactness and useit at various places.

We denote coordinates onM with the indices i, j, . . . and on N with theindices α, β, . . ..

Definition (Energy Density). For f ∈ C1(M,N ), the energy density e(f)is defined to be

e(f) := |df |2g , (2.1)

where df is the differential of f , or in local coordinates centred at a point P ,

e(f)(P ) =1

2gij(P )hαβ(f(P ))fα,if

β,j. (2.2)

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 isdefined as

E(f) =

∫Me(f)dVg. (2.3)

So E(f) is a generalisation of the Dirichlet energy functional to mapsbetween manifolds. Note that, since e(f) is non-negative, E(f) is as well.

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Definition (Harmonic Maps). A map f ∈ C2(M,N ) is said to be harmonicif f is a critical point of E(·), that is, d

dtE(ft)

∣∣t=0

= 0 for any curve ft :(−ε, ε)→ C2(M,N ) satisfying f0 = f .

The question that we are interested in answering is, given a homotopyclass [f ] ⊂ C(M,N ), does there exist a harmonic representative of [f ]?Equivalently, can every f ∈ C2(M,N ) be continuously deformed into aharmonic map?

Definition (The Tension Field). Define the tension field τ : C2(M,N ) →T (N ) by

d

dtE(ft)

∣∣∣∣t=0

= −∫M〈τ(f),

dftdt

∣∣∣∣t=0

〉dVg (2.4)

where (ft) ∈ C2(M,N ) is a family of pertubations of f in C (M,N ) suchthat f0 = f as before. The field τ is the Euler-Lagrange operator associatedto 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

τ(f)α(P ) = ∆g fα(P ) + gij(P )(ΓN )αβγ(f(P ))fβ,if

γ,j. (2.5)

Proof. For f ∈ C2(M,N ) let (ft)t∈(−ε,ε) be a smooth one parameter familyof functions such that f0 = f and ϕ =

(ddtft)t=0

. We then find

d

dt[E(ft)]t=0 =

1

2

d

dt

[∫Mgij(P )hαβ(ft(P ))fαt ,if

βt ,jdVg

]t=0

=

∫Mgij(hαβf

αt ,i∂tf

βt ,j +

1

2hαβ,γf

αt ,if

βt ,j∂tf

γt )dVg.

We can define a tangent field ξ on M by

ξi = gijfαt ,jϕβhαβ, (2.6)

where fαt ,j = fαt ;j as fαt is a scalar function with respect to M. By Green’sdivergence theorem

0 =

∫Mξi;idVg

=

∫Mgij(fαt ;jiϕ

βhαβ + fαt ,iϕβ,jhαβ + fαt ,jϕ

βhαβ,γfγt ,i)dVg.

(2.7)

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Plugging this back gives

d

dt[E(ft)]t=0 =

∫Mgij(−fαt;jiϕβhαβ−fαt,jϕβhαβ,γfγ, t,i+

1

2hαβ,γf

αt ,if

βt ,jϕ

γ)dVg.

(2.8)The first term we recognise as Laplace-Baltrami operator ∆g defined as

∆g fαt = gijfαt ;ij. (2.9)

For the remaining term we find using the symmetry of h

fαt ,jϕβhαβ,γf

γt ,i −

1

2hαβ,γf

αt ,if

βt ,jϕ

γ = ϕαfβt ,ifγt ,j

1

2(hβα,γ + hγα,β − hβγ,α)︸ ︷︷ ︸

hαδ(ΓN )δβγ

(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 thatE(f) is a critical point. If ft satisfies the heat flow

∂tft = τ(ft)

f0 = g

then we have that

d

dt(E(ft)) = −

∫M〈τ(ft), ∂tft〉dVg = −

∫M|∂tft|2dVg. (3.1)

If we can prove that ft is defined under this heat flow for all time and thatthere 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 andhence that τ(f∞) = 0. Since f∞ is homotopic to f0 via the heat flow, wewould then have that f∞ is a harmonic representative of [g].

Morally, this is the strategy which we pursue, albeit in a slightly disguisedform. In order to show the existence of a solution, we must consider the flowembedded in a Euclidean space RL. We will prove that this embedded flowadmits a unique local solution and then prove that, if the curvature of N

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is non-positive, this solution is global. We will then prove that this globalsolution admits a subsequence converging to a harmonic limit f∞ and thatthis subsequence is eventually ‘close enough’ to f∞, at which point a geodesicflow 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 localcoordinates of N . To avoid problems with local coordinates and to allow usto prove the existence of a solution, we analyse the heat flow ∂tft = τ(ft) byembedding the manifold N into some Euclidean space RL. We then provethat a solution ft ∈ C(M,RL) exists and that ft(M) ⊂ N for all timeseparately. In order to do this, we will need to reformulate ∂tft = τ(ft) interms of equations involve coordinates in RL as opposed to coordinates inN .

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 Euclideanmetric. Define u = w f , the embedding of f into RL. In order to describeτ(f) in RL, we must find an explicit expression for dw τ(f), the embeddingof the vector field τ(f) into RL, in terms of u. By compactness for smallenough δ0 > 0, the nearest point projection map π : Nδ0 → N , where

Nδ0 := y ∈ RL : d(y,N ) := infz∈N|z − y| < δ0, (4.1)

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 Euclideancoordinates the metric then reduces to the Kronecker delta δ.

Lemma 4.1. The embedded tension field dw τ(f) is the projection of ∆g uonto T (N ), ie

dw τ(f) = dπ ∆g u. (4.2)

Proof. This is a pointwise equality and so we prove it separately for everypoint P ∈ M. We choose local normal coordinates for M and N centeredat P and f(P ). In these coordinates the Christoffel symbols vanish so that

∆g ua(P ) = gijua;ij(P ) = gijua,ij(P ) (4.3)

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andτ(f)(P ) = ∆g f(P ) = gijf,ij(P ). (4.4)

By the chain rule we find

ua,ij = (u f)a,ij = wa,αfα,ij + wa,αβf

α,if

β,j. (4.5)

At P , this gives

∆g ua(P ) = gijwaα(f(P ))fα,ij(P ) + gijwa,αβ(f(P ))fα,i(P )fβ,j(P )

= waα(f(P ))τ(f)α(P ) + gijwa,αβ(f(P ))fα,i(P )fβ,j(P ).(4.6)

The first term is (dw τ(f))a, so, putting this all together, we have

∆g(u) = dw τ(f) + gijw,αβ(f(P ))fα,i(P )fβ,j(P ). (4.7)

It remains to show that gijw,αβ(f(P ))fα,i(P )fβ,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 )whose coefficients Xα, Y α with respect to the basis ∂

∂xαare constant. Then,

at f(P ),〈X, Y 〉,γ = (hαβX

αY β),γ = hαβ,γXαY β = 0 (4.8)

since the first partial derivatives of h within these coordinates are zero atf(P ). Since w is an isometric embedding, we have

0 = 〈dw X, dw Y 〉,γ = (δabwa,αw

b,βX

αY β),γ

= δab(wa,αγw

b,β + wa,αw

b,βγ)X

αY β.(4.9)

As X and Y are arbitrary vectors, the bracket itself must vanish. Permutingthe indices we find three equations

0 = δab(wa,αγw

b,β + wa,αw

b,βγ),

0 = δab(wa,γβw

b,α + wa,γw

b,αβ),

0 = δab(wa,βαw

b,γ + wa,βw

b,γα).

Now add the first two equations and subtract the third one. Using thesymmetry of δab the first term in the first equation is cancelled by the secondone in the third one. Likewise the second term in the second equation iscancelled by the first term in the third one. Finally the second term in thefirst one and the first term in the second equation are equal, so that we find

0 = δabwa,αw

b,βγ = 〈w,α, w,βγ〉. (4.10)

As wa,α spans the tangent space, this shows orthogonality.

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Next, we consider the decomposition of ∆g u into its parts parallel andnormal to T N in another way. Using local coordinates again,

∆g ua = ∆g π(u)a = (π(u)a),ijg

ij = (πa,bub,i),jg

ij

= (πa,bcub,iu

c,j + πa,bu

b,ij)g

ij

= dπ (∆g u)a + πa,bcubiucjgij

= (dπ dw τ(f))a + dπ wa,αβfα,i fβ,jg

ij + πa,bcubiucjgij

= dw τ(f)a + πa,bcubiucj,

since dπ is the orthogonal projection onto T N and w,αβfα,i f

β,jg

ij is normal toT (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

Aa(u)(∇u)(∇u) = −gijπa;bcub,iu

c,j. (4.12)

By the chain rule, ∂tuat = ∂t(w

a(ft)) = waα∂tfαt = dw ∂tft, so ut = w ft

satisfies the heat flow∂tut −∆g ut = A(ut)(∇ut,∇ut)u0 = w g

(4.13)

whenever ft satisfies ∂tft = τ(ft)

f0 = g.(4.14)

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

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

ρa,b + πa,b = δab and ρa,bc + πa,bc = 0. (4.15)

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

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

(∂t −∆g)ρ(ut)a = ρa,c (∂tu

ct −∆g u

ct)− gijρa,bcubt ,iu

ct ,j

= −ρa,cgijπc,deudt ,iuet ,j + gijπa,bcu

bt ,iu

ct ,j

= (−ρa,c + δac )gijπc,deu

dt ,iu

et ,j

= πa,cgijπc,deu

dt ,iu

et ,j.

From which we deduce (∂t−∆g)ρ(ut) = (dπA)(ut)(∇ut,∇ut). This enablesus to compute what happens to the modulus of ρ under the heat operator,

(∂t −∆g)(|ρ(ut)|2) = (∂t −∆g)(δabρ

a(ut)ρb(ut)

)= δabρ

a(ut)(∂tρ

b(ut)−∆g ρb(ut)

)+ δabρ

b(ut) (∂tρa(ut)−∆g ρ

a(ut))

− δabgij(ρa(ut),iρ

b(ut),j + .ρb(ut),iρa(ut),j

)= 2〈ρ(ut), (∂t −∆g)ρ(ut)〉 − 2|∇ρ(ut)|2

= 2〈ρ(ut), (dπ A)(ut)(∇ut,∇ut)〉 − 2|∇ρ(ut)|2

= −2|∇ρ(ut)|2

since (dπ A)(ut)(∇ut,∇ut) takes values in T (N ) and ρ(ut), as the vectorbetween ut and its nearest point on N , is perpendicular to T (N ). Sinceρ(u0) = 0, we can appeal to the maximum principle to deduce that |ρ(ut)|2 ≤0 for all t such that ut is defined. Alternatively, we can note that we havejust shown that

∂t|ρ(ut)|2 ≤ ∆g |ρ(ut)|2. (4.16)

Integrating across M and using the divergence theorem on ∆g = div gradyields

∂t

∫M|ρ(ut)|2dVg ≤ −

∫M|∇ρ(ut)|2dVg ≤ 0. (4.17)

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

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Remark. It is worth noting that, if we define E as the Dirichlet energy formaps in C1(M,RL), the operator ∆g is the tension field τ for E. Since w is anisometric embedding, u = wf , where f ∈ C1(M,N ), satisfies E(u) = E(f).Naively, we might expect to be able to use τ = ∆g to decrease E(u) and henceE(f) and to obtain our harmonic map in the limit. However, all maps withimage in RL are homotopic to constant maps, so, given convergence in thelimit, the heat flow operator ∂t −∆g will contract any initial data w g toa single point, and, in our case, forcing the image of our initial data g toleave N . The term A(u)(∇u,∇u), which we recall is normal to the surfaceof N , can be seen as a force which corrects the action of ∂t−∆g by ‘pushingback’ against it to ensure that our solution always stays on the manifold Nby negating the normal component of the flow, leaving only tangential forceswhich flow the image of g within N .

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5 The heat kernel an a Riemannian manifold

In order to construct the heat flow between the manifolds M and N , wewill use solutions of the one-dimensional heat equation onM which we thusmust understand first. For this we will define and construct a heat kernel orfundamental solution and show the required properties.

The original work was done by Minakshisundaram and Pleijel in [8] andcompleted by Minakshisundaram in [7]. A good reference for this materialis Isaac Chavel’s book [2] and an approach that is geared towards spectralgeometry may be found in [1]. Also Rosenberg’s book [9] contains a readableintroduction.

5.1 Definition Of The Heat Kernel And Its Properties

Before we actually construct a heat kernel, we will define a heat kernel by afew conditions, and then show that these already define a unique heat kernel.Afterwards, we will construct a heat kernel satisfying these conditions andthus 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 aspecific problem, i.e. we would impose that u tends to the initial data in anappropriate sense as t ↓ 0.

Now define the fundamental solution or heat kernel H.

Definition (Fundamental solution of the heat equation). Let (M, g) be aRiemannian manifold. Then H : M×M× (0,∞) 7→ R is a fundamentalsolution 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 −∆gH = 0,

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3. for every bounded continuous function f :M 7→ R and Q ∈M

limt↓0

∫MH(P,Q, t)f(P )dVg(P ) = f(Q).

Note that we impose the heat equation and the convolution with respectto 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 fas t ↓ 0.

In order to overcome this and to construct solutions for the heat equationsfrom 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)∫M

[ut−b(P )vb(P )− ut−a(P )va(P )]dVg(P )

=

∫ b

a

∫M

[ut−s(P )(Lvs)(P )− (Lut−s)(P )vs(P )] dVg(P )ds

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 )] .

Since ut−s(·) and vs(·) are C2, find over a compact manifold without boundary(Green’s theorem)∫

Mut−s(P ) ∆g vs(P )dVg(P ) = −

∫M∇ut−s(P )∇vs(P )dVg(P ).

Hence integrating the expansion beforehand over∫M dVg(P ) and

∫ ba

ds givesthe result.

With this we can prove the symmetry and uniqueness of the heat kernelwhich 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.

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Furthermore, if for f ∈ C0b (M) and F ∈ C0

b (M × (0,∞) a solutionu ∈ C0(M× (0,∞)) with C2 in space and C1 in time of

Lu = F,

limt↓0 ut(·) = f(·) uniformly,

exists, it is given by

ut(P ) =

∫MH(P,Q, t)f(Q)dVg(Q) +

∫ t

0

∫MH(P,Q, t− s)Fs(Q)dVg(Q).

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) intoDuhamel’s principle shows for [α, β] ⊂ (0, t)∫MH1(P,Q, t−β)H2(P,R, β)dVg(P ) =

∫MH1(P,Q, t−α)H2(P,R, α)dVg(P ).

In this we want to take α ↓ 0 and β ↑ t.Since M is compact and H2 is jointly continuous, there exists for every

ε > 0, some β < t such that for all β ∈ [β, t)

|H2(P,R, β)−H2(P,R, t)| < ε.

Hence

lim supβ↑t

∣∣∣∣∫MH1(P,Q, t− β)H2(P,R, β)dVg(P )−H2(Q,R, t)

∣∣∣∣≤ lim sup

β↑t

∣∣∣∣∫MH1(P,Q, t− β)H2(P,R, t)dVg(P )−H2(Q,R, t)

∣∣∣∣+ lim sup

β↑t

∣∣∣∣ε∫MH1(P,Q, t− β)dVg(P )

∣∣∣∣= 0 + ε

where we used the point-wise limit assumption of a heat kernel in the laststep. Since ε was arbitrary, this shows

limβ↑t

∫MH1(P,Q, t− β)H2(P,R, β)dVg(P ) = H2(Q,R, t)

16

and likewise

limα↓0

∫MH1(P,Q, t− α)H2(P,R, α)dVg(P ) = H1(R,Q, t).

Therefore, H1(R,Q, t) = H2(Q,R, t). By letting H1 and H2 the sameheat kernel, this shows that every heat kernel must be symmetric. Then theequality 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 limitargument for α and β the claimed result follows.

The heat kernel defines an evolution operator whose properties we collectin the following theorem.

Proposition 5.3. Suppose the compact manifold (M, d) has a heat kernelH. Define the operator family It on L2(M) for t > 0 by

It(f)(·) =

∫MH(·, Q, t)f(Q)dVg(Q).

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) hasas much regularity as H(·, ·, t). The operator family satisfies the semi-groupproperty, 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 candifferentiate twice in space and once in time by differentiating under theintegral sign showing that u satisfies the heat equation.

17

For a fixed s > 0 and f ∈ L2(M), the function Jt+s(f) satisfies the heatequation with uniform limit as t ↓ 0, so that by the previous lemma it canby 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

H the image of the unit ball in L2(M) under It is equicontinuous and thusprecompact by Arzela-Ascoli.

By the semi-group property for each f ∈ L2(M)

〈f, Ptf〉 = 〈Pt/2f, Pt/2f〉 = ‖Pt/2f‖2

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 ∫MH(P,Q, t)dVg(Q)

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 ε > 0, there exists some T > 0 so that forall t ∈ (0, T ) ∫

MH(P,Q, t)1d(P,Q)>δdVg(Q) < ε.

By compactness of M we can find for any small enough δ and ε > 0,some T such that the for all P,Q ∈M∫

MH(P,Q, t)1d(P,Q)>δdVg(Q) < ε.

Since any continuous f on M is uniformly continuous this then showsthe uniform convergence for continuous functions.

The limit in L2(M) follows now from the density of continuous functionsin L2(M) together with the previous noted fact that∫

MH(P,Q, t)dVg(Q) = 1.

18

From the positivity and the integral in the second variable we can formu-late the following corollary.

Corollary 5.4. Restricting the functions to C1(M), the operator It mapsC1(M) to C1(M) with

‖Itf‖C1 ≤ ‖f‖C1

Proof. Since f is differentiable, we can integrate by parts using the symmetryof H to control

|∇Itf(P )| =∣∣∣∣∫MH(P,Q, t)∇f(Q)dVg(Q)

∣∣∣∣ ≤ supP∈M

|∇f(P )|

since H is positive and∫MH(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 issmooth, 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 approximateheat kernel S which behaves sufficiently well as t ↓ 0. In this case we expectthe heat kernel to be concentrated around the diagonal so that we try a localapproximation.

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

(A ∗ f)(P, t) =

∫MA(P,Q, t)f(Q)dVg(Q).

Thus we expect with the heat kernel H

S ∗ f = (H +H ∗K) ∗ f

19

with

(H ∗K)(P,Q, t) =

∫ t

0

∫MH(P,R, t− s)K(R,Q, s)dVg(R)ds

so that formally

H =S

1 +K.

If the initial approximation S is good enough, then K should be small,i.e. we can define H as series by

H = S + S ∗Q with Q =∞∑n=1

(−1)nK∗n

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 diagonalM×M. In order to perform this constructionwe use spherical exponential coordinates.

For this recall the exponential map exp : TM 7→M given by

expQX = γ(1)

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

For a neighbourhood BTQM(ε) with ε > 0 the exponential map definesa diffeomorphism. By compactness of M , we can find ε > 0 such that forevery point Q ∈ M the exponential map expQ restricted to BTQM(ε) is adiffeomorphism.

Let Uε ⊂ M ×M be the neighbourhood of the diagonal in M ×Mdefined by

(P,Q) ∈ Uε if P ∈ expQ(BTQM(ε)).

By the definition of the distance all pairs (P,Q) 6∈ Uε are at least ε apart.For any (P,Q) ∈ Uε we can construct spherical exponential coordinates

around Q by using spherical coordinates (r, θ1, . . . , θm−1) on Rm ' TQM andusing the exponential map.

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

20

Gauss lemma 〈∂r, ∂θi〉 = 0 for i = 1, . . . ,m − 1. For the associated volumeelement

√gS, we find by comparison with the spherical coordinates in Rm

that

φ(P,Q) =

√gS

rm−1

defines a function φ ∈ C∞(Uε).With this setup we can formulate a local approximation Sk ∈ C∞(Uε ×

(0,∞)) of order k by

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

where G ∈ C∞(Uε × (0,∞)) is the adaption of the heat kernel in Rm, i.e.

G(P,Q, t) = (4πt)−m/2 exp

(−d

2(P,Q)

4t

),

and u0, . . . , uk ∈ C∞(Uε) are corrections. We now fix the number of correc-tions k and will show that for large enough k our procedure indeed constructsthe heat kernel H.

In order to determine the correction, understand the action of the heatoperator L in spherical harmonic coordinates at Q ∈M. For this recall thatr = d(P,Q) so that

∂tS =

(−m

2t+r2

4t2

)G (u0 + · · ·+ ukt

k) +G (u1 + · · ·+ kuktk−1)

and acting on the first variable P

∆ Sk = (∆G)(u0+· · ·+uktk)+2〈∇G,∇(u0+· · ·+uktk)〉+G∆(u0+· · ·+uktk)

where with φ′ = ∂rφ

∇G =∂r(√gS∂rG)√gs

=

(r2

4t2− m

2t− r

2t

φ′

φ

)G

and using Gauss lemma

〈∇G,∇(u0 + · · ·+ uk)〉 = − r

2t

(∂ru0 + · · ·+ ∂rukt

k)G.

21

Therefore, we find

LSk = G[(u1 + · · ·+ kukt

k−1) +r

2t

φ′

φ(u0 + · · ·+ ukt

k)

+r

t(∂ru0 + · · ·+ ∂rukt

k)− (∆u0 + · · ·+ ∆uktk)].

Hence we want to choose u0, . . . , uk such that the bracket vanishes exceptfor the highest order of t, i.e. the correction should satisfy

r

2

φ′

φu0 + r∂ru0 = 0

and for l = 1, . . . , k (l +

r

2

φ′

φ

)ul + r∂rul = ∆ul−1.

We can ensure the existence of smooth functions u0, . . . , uk satisfyingthese conditions by solving the ODEs from r = 0. In this we also imposethat u0(Q,Q) = 1 for all Q ∈M in order to match the heat operator whichwe will need in showing the limit for the convolution. With this we explicitlyfind recursively

u0(P,Q) =1√φ,

ul(P,Q) =1√φ

∫ 1

0

τ l√φ∆ul−1dτ,

where the integrand is evaluated at (rτ, θ1, . . . , θm).As φ is a positive smooth function indeed u0, . . . , uk ∈ C∞(Uε) and

LSk = G∆uktk. (5.2)

Now extend Sk ∈ C∞(Uε × (0,∞)) to Sk ∈ C∞(M×M× (0,∞)) byusing a smooth cutoff η ∈ C∞(M ×M) only depending on the distancer = d(P,Q) with η(r) ≡ 1 for r ≤ ε/2 and η(r) ≡ 0 for r ≥ ε. Then Skis indeed an approximate heat kernel for k > m/2 which we formalise asparametrix.

Definition. A function S ∈ C∞(M×M× (0,∞)) is a parametrix for theheat equation if

22

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

2. for all f ∈ Cb(M)

limt↓0

∫MS(P,Q, t)f(P )dVg(P ) = f(Q).

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

Proof. Outside Uε/2 the cutoff is smooth and the exponential exp(−r2/(4t))in G ensures that it remains smooth up to t = 0.

Inside Uε/2 we findLSl = G∆ukt

k

which converges to 0 as t 0 if k > m/2. Likewise the lth derivative existsand is bounded if k > (m/2) + 2l. (Note that for the spatial derivatives wecould just impose k > (m/2) + l.)

For the limit of convolution use the exponential coordinates φQ aroundQ to find

limt↓0

∫MSk(P,Q, t)f(P )dVg(P )

= limt↓0

∫Bε(Rm)

(4πt)−m/2 exp

(−r2

4t

)(u0 + · · ·+ ukt

k)f(φQ(x))√gφQdx

= 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 findfor any fixed time T a constant A such that for t ∈ [0, T ] and P,Q ∈M

|Kε(P,Q, t)| ≤ Atk−m/2 ≤ B

with B = AT k−m/2.

23

By induction we can then estimate

|K∗nk (P,Q, t)| ≤ ABn−1V n−1tk−(m/2)+n−1

(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

|K∗(n+1)k (P,Q, t)|

≤∫ t

0

∫M|K(P,R, t− s)K∗n(R,Q, s)|dVg(R)ds

≤∫ t

0

∫MB · ABn−1V n−1sk−(m/2)+n−1

(k − (m/2) + 1)(k − (m/2) + 2) . . . (k − (m/2) + n− 1)ds

=ABnV n

(k − (m/2) + 1)(k − (m/2) + 2) . . . (k − (m/2) + n− 1)

∫ t

0

sk−(m/2)+n−1ds

=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.Hence when k > m/2, the series

∑∞n=1 |K∗nk | converges and thus Ql =∑∞

n=1(−1)nK∗nk converges to a function in C0(M×M× [0,∞)) which con-verges uniformly to 0 as t ↓ 0.

By the same estimate as in lemma 5.5, indeed Qk ∈ C l(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 forlarge enough k we use the following lemma.

Lemma 5.6. Let F ∈ C l(M×M×[0,∞)), then Sk∗F ∈ C l(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 TPM ina smooth way with Rm and let φP : B(ε) 7→ M be expP , i.e. exponential

24

coordinates. The smooth identification can e.g. be done by parallel transporton geodesics stating at P .

Then with the cutoff η we can express the convolution as

(Sk ∗ F )(P , Q, t) =

∫ t

0

∫B(ε)

(4πs)−m/2e−|x|2/4s

(u0(P , φP (x)) + · · ·+ uk(P , φP (x)sk)η(|x|)F (φP (x), Q, t− s)dxds.

Since ∫ t

0

∫B(ε)

(4πs)−m/2e−|x|2/4sdxds <∞

we can differentiate under the integral sign and find Sk ∗ F ∈ C l(M×M×(0,∞)).

By the compactness of M×M× [0, 1], this also shows that there existsa 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 inRm (cf. [4, Section 2.3]). Hence for a small δ > 0

L(Sk ∗ F )(P,Q, t) =

∫MSk(P,R, δ)F (R,Q, t− δ)dV (R)

+

∫ t

δ

∫M

(LSk)(P, r, s)F (R,Q, t− s)dV (R)ds

+ Iδ

where

Iδ(P,Q, t) =

∫ δ

0

∫B(ε)

(4πs)−m/2e−|x|2/4s

L[(u0(P, φP (x)) + · · ·+ uk(P, φP (x)sk)η(|x|)F (φP (x), Q, t− s)

]dxds.

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

L(Sk ∗ F ) = F +Kk ∗ F.

25

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

LHk = Kk +Kk ∗∞∑n=1

(−1)nK∗nk +∞∑n=1

(−1)nK∗nt = 0.

Since Qk also tends to 0 uniformly as t ↓ 0 for k > m/2, we still have forany f ∈ Cb(M)

limt↓0

∫MH(P,Q, t)f(P )dVg(P ) = f(Q).

This shows that for Hk is a heat kernel for k > (m/2) + 4 and is inC l(M×M× (0,∞)) for k > (m/2) + 2l. By the uniqueness of the heatkernel all Hk for k > (m/2) + 4 agree and thus define the heat kernel H withinfinite regularity.

Hence we have shown the first half of our main theorem.

Theorem 5.7. Given a compact Riemannian manifold (M, g), there existsa 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]∫ t

0

∫M|H(P,Q, s)|dVg(Q)ds ≤ Ct

and ∫ t

0

∫M|∇H(P,Q, s)|dVg(Q)ds ≤ C

√t

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∫ t

0

∫M|(Sk ∗Qk)(P,Q, s)|dVg(Q)ds ≤ C1V t

and ∫ t

0

∫M|∇(Sk ∗Qk)(P,Q, s)|dVg(Q)ds ≤ C1V

√t

where V is the volume of M.

26

Since η(d(P,Q))(u0 + · · ·+ uktk) is bounded, find for some constant C2∫ t

0

∫M|Sk(P,Q, s)|dVg(Q)ds ≤ C2

∫ t

0

∫Rm

(4πt)−m/2e−|x|2/4tdxdt = C2t.

Also ∇[η(d(P,Q))(u0 + · · ·+ uktk)] is bounded, so∫ t

0

∫M|∇Sk(P,Q, s)|dVg(Q)ds ≤ C3

∫ t

0

∫Rm

(4πt)−m/2e−|x|2/4tdxdt

+ C4

∫ t

0

∫Rm

(4πt)−m/2|x|2te−|x|

2/4tdxdt

for some constants C3 and C4. The first integral is again t and for the secondintegral we find by using spherical coordinates∫

Rm(4πt)−m/2

|x|2te−|x|

2/4tdx ≤ C5t−1/2

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 thanthe 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 integraloperator. When restricted to a set of functions which stay close to the ini-tial data u0, this integral operator is a strict contraction. An application ofBanach’s Fixed Point Theorem then yields the existence of a unique localsolution. This method also yields a blow up criterion for the non existenceof global solutions, which is achieved by showing that the length of time forwhich a local solution exists is bounded below by a function of the norm ofthe initial data.

Given a heat kernelH(P,Q, t) ∈ C(M×M×(0,∞)) for the heat equationon M, we can construct an evolution operator It : C(M,RL) 7→ C(M,RL),defined by

It(u)(P ) =

∫MH(P,Q, t)u(Q) dVg(Q),

which satisfies the heat equation for any initial data u0 ∈ C(M,RL). ViaDuhamel’s principle, we can use It to convert the equation (4.13) into anintegral equation as follows: Multiplying (4.13) by the integrating factorIt−s, we obtain

d

ds(It−sus) = It−sA(us)(∇us,∇us).

Integrating with respect to s from 0 to t then leaves us with

ut = Itu0 +

∫ t

0

It−sA(us)(∇us,∇us) ds (6.1)

as required. Hence, to find a solution of (4.13), it suffices to find a fixedpoint of the map

T : v 7→ Itu0 +

∫ t

0

It−sA(vs)(∇vs,∇vs) ds.

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

28

u0 is only assumed to be in C(M,RL) and so the differentiability of Tv att = 0 is not guaranteed. This issue is easily remedied by the following lemma,which shows that we can always assume that u0 ∈ C∞(M;RL).

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 ε > 0 such that, for t ≤ ε,(Itv)(M) is contained within the domain of definition of π, the nearest pointprojection operator from the tubular neighbourhood Nδ0 to N . Hence, themap t 7→ (π It)v describes a homotopy within C(M,N ) between v andv := (π Iε)v. The fact that v ∈ C∞(M,N ) now follows from the fact thatπ is a smooth map combined with Proposition 5.3.

DefineXε := L∞([0, ε), C1(M,RL)). (6.2)

It is clear that, when endowed with the norm

‖u‖Xε = supt∈[0,ε)

‖ut‖C1(M,RL) ,

Xε is a Banach Space.

Theorem 6.2. The integral equation (6.1) admits a unique local solution

u ∈⋃

0≤ε<Tmax

Xε,

where [0, Tmax) is the maximal interval of existence of the solution. Moreover,if Tmax <∞ then limt↑Tmax ‖∇ut‖C(M) =∞.

Proof. Throughout this section C will denote a positive constant which,whilst its value may change line by line, depends only on M and δ0.

Define T : Xε → Xε by

Tv = Itu0 +

∫ t

0

It−sA(vs)(∇vs,∇vs)ds. (6.3)

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

Bδ := v ∈ Xε : ‖v − Itu0‖Xε ≤ δ. (6.4)

29

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

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

Aa(v)(∇v)(∇v) = −gijπa;bcvb,iv

c,j,

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

|Tv − Itu0|(P ) =

∣∣∣∣∫ t

0

∫MH(P,Q, t− s)A(vs)(∇vs,∇vs)(Q)dVg(Q)ds

∣∣∣∣≤ sup

s∈[0,t]

‖A(vs)(∇vs,∇vs)‖C(M)

∫ t

0

∫M|H(P,Q, t− s)|dVg(Q)

≤ Cε sups∈[0,t]

‖∇vs‖2C(M) ≤ Cε ‖v‖2

Xε,

and similarly that

|∇(Tv − Itu0)|(P ) =

∣∣∣∣∫ t

0

∫M∇H(P,Q, t− s)A(vs)(∇vs,∇vs)(Q)dVg(Q)ds

∣∣∣∣≤ C√ε ‖v‖2

Xε.

It follows that‖Tv − Itu0‖Xε ≤ C

√ε ‖v‖2

Xε.

Note that, for v ∈ Bδ,

‖v‖Xε ≤ ‖Itu0‖Xε + ‖v − Itu0‖Xε ≤ 1 + ‖Itu0‖X1,

and so, upon taking√ε =

δ

C(1 + ‖Itu0‖X1

)2

we are able to guarantee that T (Bδ) ⊂ (Bδ).

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

‖Tu− Tv‖Xε ≤ λ ‖u− v‖Xε for all u, v ∈ Bδ,

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 onM, Aa(v)(∇v,∇v) = −gijπa,bcvb,ivc,j we have that

Aa(u)(∇u,∇u)− Aa(v)(∇v,∇v) =− gij(πa,bc(u)ub,iuc,j + πa,bc(v)vb,iv

c,j)

=− gij(πa,bc(u)(ub,iuc,j − vb,ivc,j)

− (πa,bc(u)− πa,bc(v))vb,ivc,j).

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

gijπa,bc(u)(ub,i + vb,i)(uc,j − vc,j) = πa,bc(u)(gij(ub,iu

c,j − vb,ivc,j)

+ gij(vb,iuc,j − ub,ivc,j))

= gijπa,bc(u)(ub,iuc,j − vb,ivc,j)

+ gijπa,bc(u)vb,iuc,j − gjiπa,cb(u)vc,ju

b,i

= gijπa,bc(u)(ub,iuc,j − vb,ivc,j).

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

|πa,bc(u)− πa,bc(v)| ≤ supx∈Nδ0

(|πa,bcd|(x)) |ud − vd|. (6.5)

N is compact, so Nδ0 is as well, so |πa| and all of its derivatives are boundedby 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

|Tut − Tvt| ≤∫ t

0

|It−s (A(us)(∇us,∇us)− A(vs)(∇vs,∇vs)| ds

≤ C sups∈[0,t]

‖A(vs)(∇vs,∇vs)− A(us)(∇us,∇us)‖C(M)

∫ t

0

|It−s| ds

≤ Cε sups∈[0,t]

((1 + ‖∇us‖C(M) + ‖∇vs‖C(M)

)‖∇us −∇vs‖C(M)

),

31

and similarly that

|∇Tut−∇Tvt| ≤ C√ε sups∈[0,t]

((1 + ‖∇us‖C(M) + ‖∇vs‖C(M)

)‖∇us −∇vs‖C(M)

).

Taking the supremum in P and t and combining these two inequalities leavesus with

‖Tv − Tu‖Xε ≤ C√ε(1 + ‖u‖Xε + ‖v‖Xε

)‖u− v‖Xε

≤ C√ε(1 + ‖Itu0‖X1

)‖u− v‖Xε ,

Hence, by taking

√ε = min

δ

C(1 + ‖Itu0‖X1

)2 ,1

C(1 + ‖Itu0‖X1

) ,we can guarantee that T is a strict contraction from Bδ to Bδ with contractionconstant λ ≤ 1

2.

By Banach’s Fixed Point Theorem, it follows that T admits a fixed pointu ∈ Bδ and so, following on from the discussion in the introduction to thissection, that (4.13) is guaranteed a short time solution valid for a time periodof length at least ε. From Proposition 5.3 and Equation (6.1) we can thendeduce that ut ∈ C∞(M × (0,∞),RL) 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 maximalinterval of existence [0, Tmax) and that ‖ut‖XTmax

<∞. Let t0 ∈ [0, Tmax) andconsider the new heat flow

∂tvt −∆g vt = A(vt)(∇vt,∇vt)v0 = ut0 .

(6.6)

By the discussion following Lemma 6.4, the equation (6.6) admits a uniquelocal solution, valid for a time period of length at least ε, where

√ε = min

δ

C(

1 + ‖Itut0‖Xt0+1

)2 ,1

C(

1 + ‖Itut0‖Xt0+1

) .

32

By Corollary 5.4, we have that ‖Itut0‖Xt0+1≤ ‖ut0‖C1(M,RL). This implies

that ε is bounded uniformly away from 0 for t0 ∈ [0, Tmax), by ε0, say. Takingt0 to be such that Tmax−t0 < ε0, then, we see that ut can be continued in a waywhich satisfies equation 4.13 beyond t = Tmax by defining ut0+s = vs. Thiscontradicts the maximality of the interval [0, Tmax) and hence demonstratesthat, if Tmax < ∞, we must have that ‖ut‖C1(M,RL) → ∞ as t ↑ Tmax. Sinceut takes values within the compact manifold w(N ), ‖ut‖C(M,RL) is uniformlybounded. Hence, we must in fact have that ‖∇ut‖C(M,RL) →∞ as t ↑ Tmax.

7 Global solution to the heat flow

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

Theorem 7.1. If M is compact and N has non-positive sectional curvaturethen the heat flow

∂tft = τ(ft),

f(·, 0) = f0,(7.1)

admits a solution ft ∈ C∞(M,N ) valid for all time. Further we can finda 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 esti-mates 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 thatwe have convergence and also to show that the limiting function f∞ is in facta harmonic map. Rather that the entire heat flow (ft) converges, however, wewill only show that a subsequence of samples (ftk) converges as tk ↑ ∞. Wewill then use the compactness of M to argue that eventually ftk is so closeto f∞ that each point ftk(P ) is connected to f∞(P ) by a unique geodesic andhence 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 theo-rem 7.8. Recall, that the tension field τ(f) is the gradient of E(f) in the sense

33

that∫M〈τ(f), dft

dt

∣∣t=0〉dVg gives the directional derivative of E(·) at f in the di-

rection dftdt

∣∣t=0

. If ft satisfies the heat flow ∂tft = τ(ft), we will have

d

dt(E(ft)) = −

∫M〈τ(ft), ∂tft〉dVg

= −∫M|∂tft|2dVg

= −∫M|τ(ft)|2dVg ≤ 0 (7.2)

and so E(ft) will decrease as t→∞. Furthermore E(ft) is strictly decreasingunless τ(ft) = 0 (whence ft is harmonic). Since the energy is a quadraticform, E(ft) is bounded below. We will, however, have to work a little harderto show that limt→∞E(ft) = inft∈[0,∞) E(ft).

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

0 ≤ E(ft) = E(f0)−∫ t

0

‖τ(fs)‖2L2(M)ds ≤ E(f0), 0 < t < T. (7.3)

Proof. Integrating (7.2) yields the desired inequality.

We prove a first estimate.

Proposition 7.3 (Bochner Identity). If ft ∈ C3,1(M× [0, T ),N ) satisfiesthe heat flow then

(∂t −∆g)e(ft) + |∇df |2 + (RicM)ijfαt,ifβt,jhαβ = (RN )αβγδf

αt,if

γt,jf

βt,kf

δt,lg

ijgkl

where ∇ is the covariant derivative on T ∗M⊗ f−1TN , RicM is the Riccitensor of M and RN is the Riemann tensor of N .

Taking a stationary ft reduces to the Bochner inequality for harmonicmaps.

Proof. We prove this pointwise for every P ∈M and t ∈ [0, T ). For this wechoose local normal coordinates on M and N centered at P and ft(P ).

34

In these we find

(∆g−∂t)e(ft) =1

2(∆g−∂t)gijhαβfαt ,if

βt ,j

= gijgkl(1

2hαβ,γδf

γt ,kf

δt ,lf

αt ,if

βt ,j + hαβf

αt ;ikf

βt ;jl)

+ hαβfαt ,if

βt ;jkl − hαβfαt ,i(∂tf

βt ),j).

By the Ricci identity

fβt ;jkl = fβt ;klj + (RM)lpkjgpqfβt ,q.

As ft satisfies the heat equation we find

gijgkl(hαβfαt ,if

βt ;jkl − hαβfαt ,i(∂tf

βt ),j)

= gijgklhαβfαt ,i(−(ΓN )βγδf

γt ,kf

δt ,l),j

= −gijgklhαβ(ΓN )βγδ,εfαt ,if

γt ,kf

δt ,lf

εt ,j.

Hence we arrive at

(∂t −∆g)e(ft) + |∇dft|2 + (RM)ijfαt ,ifβt ,jhαβ

= gijgklfαt ,ifβt ,jf

γt ,kf

δt ,l((Γ

N )εγδ,βhαε −1

2hαβ,γδ). (7.4)

Recall the Christoffel symbols

(ΓN )αβγ =1

2hαδ(hβδ,γ + hγδ,β − hβγ,δ).

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

(RN )αβγδ = (ΓN )αβδ,γ − (ΓN )αβγ,δ

=1

2hαε(hδε,βγ + hβγ,εδ − hβδ,εγ − hγε,βδ).

On the other hand for our bracket in (7.4)

(ΓN )εγδ,βhαε −1

2hαβ,γδ =

1

2(hγα,δβ + hδα,γβ − hαβ,γδ − hγδ,αβ) = Rαγβδ

which proves our result.

35

Corollary 7.4. Suppose that M is compact and that N has non-positivesectional curvature, that is, KN ≤ 0. If ft solves the heat equation then wehave

(∂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 HeatEquation). Let v ∈ C∞(BR(x0) × [t0 − R2, t0]) be a non-negative functionsatisfying

∂v

∂t−∆v ≤ C1v,

on the cylinder BR(x0) × (t0 − R2, t0], where C1 ∈ R. Then there exists aconstant C2 such that

v(x0, t0) ≤ C2R−(m+2)

∫ t0

t0−R2

∫BR(x0)

v(y, s)dVgds.

For completeness we include the following result, which we do not use inthe proof of Theorem 7.1, however it does show that any stationary point ofthe 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 curvaturethen d2

dt2E(ft) ≥ 0.

Proof. Denote by DN

dtthe covariant derivative (considering fβt,j as a tangent

vector in N ):

d

dte(ft) = hαβft

α,i

(DN

dtftβ,j

)gij

d2

dt2e(ft) = hαβft

α,i

(DN

dt

DN

dtftβ,j

)gij + hαβ

(DN

dtftα,i

)(DN

dtftβ,j

)gij.

36

As the connection is torsion-free (where ∇N denotes the covariant derivativeon N ):

DN

dtftα,i = ∂tf

βt ∇Nβ fαt ,i

= ∂tfβt

(∂ft

α,i

∂yβ+ Γαγβf

γt ,i

)= ∇Ni ∂tfαt ,

and using the Ricci identity,

DN

dt∇Ni ∂tfαt = ∇Ni

DN

dt∂tf

αt + (RN )αβγδ∂tf

γt f

δt ,i∂tf

βt .

Furthermore,

∇Ni (hαβftα,jg

ij) = hαβ∆gfαt + hαβ(ΓN )αγδf

γt ,jf

δt ,ig

ij

= τ(f)β.

Hence, integrating by parts we find (since (RN )αβ[γδ] = 0):

1

2

d2E(ft)

dt2=

∫Mhαβ

(∇Ni ∂tfαt

) (∇Nj ∂tf

βt

)gijdVg

−∫M

(RN )αβγδfαt ,if

γt ,j∂tf

βt,i∂tf

δt g

ijdVg.

Therefore, if N has non-positive sectional curvature, we have d2E(ft)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 hasnon-negative sectional curvature. We have the following a priori estimate:

‖∇ft‖C0(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:

e(ft0)(x0) ≤ CR−(m+2)

∫ t0

t0−R2

∫M

e(ft)dVg dt

= CR−(m+2)

∫ t0

t0−R2

E(ft)dt

≤ CR−mE(f0),

by the energy inequality, Lemma 7.2. So |∇ft| is uniformly bounded onM×(1, T ). To bound the gradient onM×[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,

e(ft)(x) ≤ ‖f0‖C0(M)eCt ≤ eC‖f0‖C0(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 curvaturethen 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 boundedon M× (0, T ). Since, by Theorem 6.2 we have that the gradient blows upas 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 7.7to e(∇m−1ft), m = 2, ... we have

supt∈[0,∞)

‖∇mft‖C0(M) ≤ C(m,M,N , u0), ∀m ≥ 1 (7.6)

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, thelimit f∞ is harmonic.

Proof. By the energy inequality, Lemma 7.2, we have that for any t > 0,∫ t

0

∫M|∂tft|2 dVgdt ≤ E(f0) <∞.

From this, we may deduce that

limt↑∞

∫ t

t−2

∫M|∂tft|2 dVgdt = lim

t↑∞

(∫ t

0

∫M|∂tft|2 dVgdt−

∫ t−2

0

∫M|∂tft|2 dVgdt

)= 0.

Since

(∂t −∆g) |∂tft|2 = − |∂tft|+RN (ft)(∇ft, ∂tft,∇ft, ∂tft) ≤ 0,

we may apply Lemma 7.5 to find that for any α ∈ (0, 1),

‖∂tft‖Cα(M×[t−1,t]) ≤ C(α) ‖∂tft‖L2(M×[t−2,t]) → 0, (7.7)

as t → ∞. This implies that, for all t, ft ∈ Cα(M) with Holder constantuniformly bounded and so by Arzela-Ascola we find some sequence tk ↑ ∞such that ∂tftk → 0, by (7.7), in Cα(M). So we have that f∞ := limk→∞ ftkis harmonic. It can be shown by a further application of Arzela-Ascoli usingthe 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 thatd(ftk(P ), f∞(P )) is less than the radius of injectivity for all P ∈ M. Thenthe 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 thesehomotopies 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 thecase 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 thatthe proof of long time existence presented here is no longer valid. Struwedeals with this by showing that this type of singularity can only occur on afinite set and that, once these singularities are accounted for, the heat flowcan still be continued. Heat flow methods have found applications in manyother areas of geometry, most notably in Richard Hamilton’s programme forthe use of Ricci Flow, where a heat operator is applied to the curvature ofthe manifold, to solve the Thurston Geometrization conjecture which wasfamously completed by Perelman in 2003.

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[2] Isaac Chavel. Eigenvalues in Riemannian geometry, volume 115 of Pureand Applied Mathematics. Academic Press Inc., Orlando, FL, 1984. In-cluding a chapter by Burton Randol, With an appendix by Jozef Dodz-iuk.

[3] James Eells, Jr. and J. H. Sampson. Harmonic mappings of Riemannianmanifolds. Amer. J. Math., 86:109–160, 1964.

[4] Lawrence C. Evans. Partial differential equations, volume 19 of GraduateStudies in Mathematics. American Mathematical Society, Providence,RI, second edition, 2010.

[5] Robert Hardt and Michael Wolf, editors. Nonlinear partial differentialequations in differential geometry, volume 2 of IAS/Park City Mathe-matics Series. American Mathematical Society, Providence, RI, 1996.Lecture notes from the Summer School held in Park City, Utah, June20–July 12, 1992.

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[6] Fanghua Lin and Changyou Wang. The analysis of harmonic maps andtheir heat flows. World Scientific Publishing Co. Pte. Ltd., Hackensack,NJ, 2008.

[7] S. Minakshisundaram. Eigenfunctions on Riemannian manifolds. J.Indian Math. Soc. (N.S.), 17:159–165 (1954), 1953.

[8] S. Minakshisundaram and A. Pleijel. Some properties of the eigenfunc-tions of the Laplace-operator on Riemannian manifolds. Canadian J.Math., 1:242–256, 1949.

[9] Steven Rosenberg. The Laplacian on a Riemannian manifold, volume 31of London Mathematical Society Student Texts. Cambridge UniversityPress, Cambridge, 1997. An introduction to analysis on manifolds.

[10] Michael Struwe. On the evolution of harmonic mappings of Riemanniansurfaces. Comment. Math. Helv., 60(4):558–581, 1985.

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