# Spectral and Hodge theory of “Witt” incomplete cusp edge spaces

### Jesse Gell-Redman

The University of Melbourne, Parkville, Australia### Jan Swoboda

Universität Heidelberg, Germany

## Abstract

Incomplete cusp edges model the behavior of the Weil–Petersson metric on the compactified Riemann moduli space near the interior of a divisor. Assuming such a space is Witt, we construct a fundamental solution to the heat equation, and using a precise description of its asymptotic behavior at the singular set, we prove that the Hodge–Laplacian on differential forms is essentially self-adjoint, with discrete spectrum satisfying Weyl asymptotics. We go on to prove bounds on the growth of $L_{2}$-harmonic forms at the singular set and to prove a Hodge theorem, namely that the space of $L_{2}$-harmonic forms is naturally isomorphic to the middle-perversity intersection cohomology. Moreover, we develop an asymptotic expansion for the heat trace near $t=0$.

## Cite this article

Jesse Gell-Redman, Jan Swoboda, Spectral and Hodge theory of “Witt” incomplete cusp edge spaces. Comment. Math. Helv. 94 (2019), no. 4, pp. 701–765

DOI 10.4171/CMH/472