Analytic Torsion on Manifolds with Edges

TL;DR

This paper investigates the properties of analytic torsion on odd-dimensional incomplete Riemannian spaces with edge singularities, establishing conditions for its well-definedness and independence from metric choices.

Contribution

It proves the holomorphicity of the torsion zeta function near zero and shows torsion's dependence only on asymptotic structure, with independence in odd dimensions.

Findings

Torsion zeta function is holomorphic near s=0 for admissible edge metrics.

Analytic torsion depends only on the asymptotic structure of the metric near singularities.

In odd dimensions, the analytic torsion is independent of the choice of admissible edge metric.

Abstract

Let (M,g) be an odd-dimensional incomplete compact Riemannian singular space with a simple edge singularity. We study the analytic torsion on M, and in particular consider how it depends on the metric g. If g is an admissible edge metric, we prove that the torsion zeta function is holomorphic near s = 0, hence the torsion is well-defined, but possibly depends on g. In general dimensions, we prove that the analytic torsion depends only on the asymptotic structure of g near the singular stratum of M; when the dimension of the edge is odd, we prove that the analytic torsion is independent of the choice of admissible edge metric. The main tool is the construction, via the methodology of geometric microlocal analysis, of the heat kernel for the Friedrichs extension of the Hodge Laplacian in all degrees. In this way we obtain detailed asymptotics of this heat kernel and its trace.