theory on moduli space, our main technical contribution is the construction and detailed description of the heat kernel $H = \exp(- t \Delta^{g_{{\mathrm{ice}}}})$. Indeed, our approach to establishing Theorem \[thm:essentiallyselfadjoint\] (which justifies the use of the word ‘the’ in the previous sentence) and Theorem \[thm:hodge\], is to develop in Theorem \[thm:heatkernel\] below a precise understanding of the behavior of a fundamental solution to the heat equation, which we only conclude is *the* heat kernel after using it to prove Theorem \[thm:essentiallyselfadjoint\]; we establish asymptotic expressions for it at the singular set, uniformly down to time $t = 0$, obtaining in particular in Corollary \[thm:heat-trace\], an asymptotic formula for its trace (which has potential applications to index theory, since our method for analyzing $\Delta^{g_{{\mathrm{ice}}}}$ may be used for other natural elliptic differential operators on these spaces as well) and fine mapping properties of $\Delta^{g_{{\mathrm{ice}}}}$ which allow us to analyze its kernel, i.e.harmonic forms. This is all described in detail in Section \[sec:proofs\].

Essential self-adjointness of a differential operator $P$ is typically a statement about the decay of $L^2$ sections $u$ for which $P u \in
L^2$. (Here the derivative is taken in the distributional sense.) The set of such sections is 