A lower bound for the heat kernel

✍ Scribed by Jeff Cheeger; Shing-Tung Yau

• Publisher: John Wiley and Sons
• Year: 1981
• Tongue: English
• Weight: 579 KB
• Volume: 34
• Category: Article
• ISSN: 0010-3640
• DOI: 10.1002/cpa.3160340404

✦ Synopsis

Institute for Advanced Study

0. Introduction

In this note we give an estimate from below for the fundamental solution of the heat equation on an open metric ball BR(x), of radius R, in a Riemannian manifold M". We assume that, for all r < R, B,(x) is compact. The estimate is in the form of a comparison with the fundamental solution on a "model" space Gx", the mean curvatures of whose distance spheres are greater than the corresponding mean curvatures in M". By a standard argument, the comparison of mean curvatures can in turn be reduced to a comparison of Ricci curvatures in radial directions.

We note that in [9] Debiard, Gaveau, and Mazet have proved a particular case of this estimate. They consider a normal coordinate ball with Dirichlet boundary conditions and use a lower bound on the sectional curvature to compare with a model space of constant curvature. The argument which allows one to avoid the assumption on the injectivity radius is essentially that of [6]; see also [ 151. We refer to [4] and [8] for some earlier results which are related to those of the present paper.

1. Basic Facts about the Heat Kernel of a General Riemannian Manifold

Recall that if Q is an arbitrary (perhaps incomplete) Riemannian manifold without boundary, then the domain 9 ( d ) of the exterior derivative, d, on functions, is defined as consisting of those f E C"(Q) for which Since the operator 6 = -* d + on one-forms, satisfies (df,,) = (f,&), if f, a, are smooth and one of them is compactly supported, d has a closure d, whose domain is denoted by H'(Q) c L'(S2). The operator d, is defined to be the restriction of d to CF(S2) and the domain of its closure 2, is denoted by