Short-time heat content asymptotics via the wave and eikonal equations

In this short paper, we derive an alternative proof for some known [van den Berg & Gilkey 2015] short-time asymptotics of the heat content in compact full-dimensional submanifolds $S$ with smooth boundary. This includes formulae like \begin{equation*} \int_{S} \exp(t\Delta)\left( f \mathbb 1_S\right), $\mathrm{d}$x = \int_S f ,$\mathrm{d}$x - \sqrt{\frac{t}{\pi}} \int_{\partial S} f ,$\mathrm{d}$A + o(\sqrt t),\quad t \rightarrow 0,. \end{equation*} and (partially new) explicit expressions for similar expansions involving other powers of $\sqrt{t}$. By the same method, we also obtain short-time asymptotics of ${\int }_S\exp (t^m{\mathrm{\Delta }}^m)\left(f{\mathbb{1}}_S\right)\mathrm{d}x$, $m \in \mathbb N$,andmoregenerallyforone-parameterfamiliesofoperators$t \mapsto k(\sqrt{-t\Delta})$definedbyanevenSchwartzfunction$k$.