the Deligne-Mumford compactifications (see e.g. [@Harris-Morrison; @Wolpert-WP-geometry]). Then the de Rham cohomology spaces $H_{dR}({\overline{\mathcal{M}}}_{1,1})$ are naturally isomorphic to $\mathcal{H}^*_{L^2}({\overline{\mathcal{M}}}_{1,1}, g_{WP})$, and the same holds for ${\overline{\mathcal{M}}}_{0, 4}$.

We discuss the proof at the end of Section \[sec:hodge-theory\], though this is really a direct application of our results together with the recent work on the structure of the Weil-Petersson metric near a divisor in [@MS2015] and [@MZ2015].

This article is partly motivated by Ji, Mazzeo, Müller, and Vasy’s work [@JMMV2014] on the spectral theory of the (scalar) Laplace-Beltrami operator on the Riemann moduli spaces $\mathcal M_{g}$, for which it was shown by methods different from ours that it is essentially self-adjoint and its eigenvalues satisfy a Weyl asymptotic formula. Here they analyze incomplete cusp edge spaces with normal crossings, and find in particular that the value $k = 3$ in is critical; indeed *for values $k < 3$ one does not expect self-adjointness*. It would be interesting (though more complex) to find a parametrization of the space of closed extensions of incomplete cusp edge Laplacians with $k < 3$, which is expected to be infinite dimensional, e.g. by [@ALMPII].

In contrast with [@JMMV2014], since our eventual goal is Hodge and index 