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Heat Flow Methods in Geometry and Topology

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Heat Flow Methods in Geometry and Topology 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 jrfj dx, defined for functions f 2 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 func- tional E(f) to functions f 2 C1(M; N ) between abstract Riemannian man- ifolds (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 2 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 ! 1, a harmonic map occurs in 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 flow in 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 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 2 M, like Rm, and the relation is smooth. That is, for all P 2 M, there is a smooth diffeomorphism ' between a neighbourhood of P and an open subset of Rm. We refer to the Euclidean coordinates of '(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 ' 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 gjp 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 2 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 r 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. rirjf = rjrif for a scalar function f, and is compatible with the metric, that is, rigjk = 0. We always use this connection and denote X;j = rjX: (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 p f(P )dVg(P ) = f('(x)) gdx M V 5 p 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 p X @i( g@if) ∆ f = p 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 2 C1(M; N ), the energy density e(f) is defined to be 2 e(f) := jdfjg ; (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.
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