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arXiv · 1509.06359

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

Jesse Gell-Redman·Jan Swoboda
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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 $t = 0$.

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Jesse Gell-Redman, Jan Swoboda. 2015-09-21. Spectral and Hodge theory of `Witt' incomplete cusp edge spaces. https://arxiv.org/abs/1509.06359

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