Spectral and Hodge theory of `Witt' incomplete cusp edge spaces

TL;DR

This paper develops spectral and Hodge theory for Witt incomplete cusp edge spaces, establishing self-adjointness, spectral properties, and a Hodge theorem linking harmonic forms to intersection cohomology.

Contribution

It constructs a fundamental solution to the heat equation and proves key spectral and geometric properties for these singular spaces, extending classical Hodge theory.

Findings

Hodge-Laplacian is essentially self-adjoint with discrete spectrum

Established bounds on growth of $L^2$-harmonic forms at singularities

Proved a Hodge theorem relating harmonic forms to intersection cohomology

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 $t = 0$.