the relationship between the various boundary hypersurfaces (codimension one boundary faces) of ${M^2_{\operatorname{heat}}}$ and those of $M \times M \times [0, \infty)_t$. The upshot is that the heat kernel $H$, which lives a priori on the latter space, pulls back via $\beta$ to be “nice” (precisely to be polyhomogeneous, see Appendix \[sec:mwc\]) on ${M^2_{\operatorname{heat}}}$. In fact, in Section \[sec:heat-kernel\] *we will construct a parametrix $K$ for the heat equation directly on ${M^2_{\operatorname{heat}}}$*. To obtain the actual heat kernel $H$ we use a Neumann series argument to iterate away the error.

The latter process builds on what is now a substantial body of work on analysis (in particular the structure of heat kernels) on singular and non-compact Riemannian spaces, going back at least to the work of Cheeger on manifolds with conical singularities [@C1979; @C1980; @C1983]. Our approach here is more closely related to Melrose’s geometric microlocal analysis on asymptotically cylindrical manifolds [@tapsit] (a non-compact example) and Mooers’ paper [@Mooers-Edith] on manifolds with conical singularities (an incomplete, singular example). The general procedure, which one sees in both the parabolic and elliptic settings, is to express the relevant differential operator as an element in the universal enveloping algebra of a Lie algebra 