Heat flow in Riemannian manifolds with non-negative Ricci curvature

TL;DR

This paper investigates heat flow in non-compact Riemannian manifolds with non-negative Ricci curvature, providing conditions for finite heat content and bounds on heat transfer for open sets.

Contribution

It establishes necessary and sufficient conditions for infinite measure sets to have finite heat content and derives bounds on heat content and heat loss.

Findings

Finite heat content conditions for infinite measure sets

Upper and lower bounds for heat content in manifolds

Two-sided bounds for heat loss when measure is finite

Abstract

Let $\Omega$ be an open set in a geodesically complete, non-compact, $m$-dimen-sional Riemannian manifold $M$ with non-negative Ricci curvature, and without boundary. We study the heat flow from $\Omega$ into $M-\Omega$ if the initial temperature distribution is the characteristic function of $\Omega$. We obtain a necessary and sufficient condition which ensures that an open set $\Omega$ with infinite measure has finite heat content for all $t>0$. We also obtain upper and lower bounds for the heat content of $\Omega$ in $M$. Two-sided bounds are obtained for the heat loss of $\Omega$ in $M$ if the measure of $\Omega$ is finite.