Heat content in non-compact Riemannian manifolds

TL;DR

This paper investigates the heat content in open sets within non-compact Riemannian manifolds, establishing conditions under which finite heat content at one time implies finiteness at all times, with bounds provided for such heat content.

Contribution

It proves that in non-compact Riemannian manifolds with Gaussian heat kernel bounds, finite heat content at some time implies finiteness at all times, with comparable bounds derived.

Findings

Finite heat content at some time implies finiteness at all times.

Established two-sided bounds for heat content in non-compact manifolds.

Results apply to open sets with infinite measure under certain conditions.

Abstract

Let $\Omega$ be an open set in a complete, smooth, non-compact, $m$-dimensional Riemannian manifold $M$ without boundary, where $M$ satisfies a two-sided Li-Yau gaussian heat kernel bound. It is shown that if $\Omega$ has infinite measure, and if $\Omega$ has finite heat content $H_{\Omega}(T)$ for some $T>0$, then $H_{\Omega}(t)<\infty$ for all $t>0$. Comparable two-sided bounds for $H_{\Omega}(t)$ are obtained for such $\Omega$.