Heat Kernel for Open Manifolds

TL;DR

This paper extends the understanding of heat kernels on open manifolds by proving a key property for complete manifolds with Ricci curvature bounds and providing an integral representation of the heat kernel of degree k.

Contribution

It offers a proof of a heat kernel property for complete manifolds with Ricci curvature bounds and derives an integral representation for the heat kernel of degree k.

Findings

Proved a property of the heat kernel for complete manifolds with Ricci curvature bounded below.

Derived an integral representation of the heat kernel of degree k.

Extended previous results on heat kernels to a broader class of manifolds.

Abstract

In a 1991 paper by Buttig and Eichhorn, the existence and uniqueness of a differential forms heat kernel on open manifolds of bounded geometry was proven. In that paper, it was shown that the heat kernel obeyed certain properties, one of which was a relationship between the derivative of heat kernel of different degrees. We will give a proof of this condition for complete manifolds with Ricci curvature bounded below, and then use it to give an integral representation of the heat kernel of degree $k$.