Fundamental properties of manifolds

The goal of this chapter is to obtain information about the heat kernel of a complete Riemannian manifold. There is a tremendous literature on this subject, much of which concerns the asymptotic form of the heat kernel K(t, x, y) as t → 0; we, however, shall be mainly interested in finding uniform upper and lower bounds over the whole range of t, x, y. For manifolds of non-negative Ricci curvature this problem is now largely solved as a result of the efforts of Li and Yau, whose work we shall describe below; see Corollary 5.3.6 and Theorems 5.5.6 and 5.6.3.

If one merely assumes that the Ricci curvature of the manifold is bounded below by a negative constant, the above methods can still be applied but give a much less complete picture. Indeed even for hyperbolic space and its quotients by Kleinian groups the variety of phenomena which can occur is vast and only partly understood. In Section 5.7 we give a brief summary of some recent results, without proofs.

We start with a brief introduction to manifold theory in order to fix notation. Let M be an n-dimensional (connected) manifold with tangent space TM and cotangent space T*M. Smooth sections of TM are called vector fields and smooth sections of T*M are called forms.