Heat flow in Riemannian manifolds with non-negative Ricci curvature

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.