---
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 $t = 0$.'
address:
- 'Department of Mathematics, Johns Hopkins University'
- Mathematisches Institut der LMU München
author:
- 'Jesse Gell-Redman'
- Jan Swoboda
title: 'Spectral and Hodge theory of ‘Witt’ incomplete cusp edge spaces'
---

Introduction
============

On a compact manifold $M$ with boundary ${\partial}M$ which is the total space of a fiber bundle $$\label{eq:boundary-fibration}
Z \hookrightarrow {\partial}M {\xrightarrow{\phantom{x} \pi \phantom{x}}} Y,$$ with $Z, Y$ closed manifolds, an *incomplete cusp edge metric* $g_{ice}$ is, roughly speaking, a smooth Riemannian metric on the 