Heat content in non-compact Riemannian manifolds

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 $\Omega$.