Heat content in non-compact Riemannian manifolds

Let $\mathrm{\Omega }$ 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 $\mathrm{\Omega }$ has infinite measure, and if $\mathrm{\Omega }$ has finite heat content $H_{\mathrm{\Omega }}(T)$ for some $T>0$, then $H_{\mathrm{\Omega }}(t)<\mathrm{\infty }$ for all $t>0$. Comparable two-sided bounds for $H_{\mathrm{\Omega }}(t)$ are obtained for such $\mathrm{\Omega }$.