Spectral geometry on manifolds with fibred boundary metrics II: heat kernel asymptotics

TL;DR

This paper advances the spectral analysis of manifolds with fibred boundary metrics by constructing the heat kernel for the Hodge Laplacian, facilitating future studies of torsion, eta-invariants, and index theorems.

Contribution

It generalizes previous work by constructing the heat kernel as a polyhomogeneous conormal distribution on manifolds with corners for fibred boundary metrics.

Findings

Constructed the heat kernel for the Hodge Laplacian on fibred boundary manifolds.

Provided a framework for analyzing spectral invariants like Ray-Singer torsion and eta-invariants.

Extended the analysis to a broader class of complete manifolds with fibred boundary metrics.

Abstract

In this paper we continue the analysis of spectral problems in the setting of complete manifolds with fibred boundary metrics, also referred to as $\phi$-metrics, as initiated in our previous work. We consider the Hodge Laplacian for a $\phi$-metric and construct the corresponding heat kernel as a polyhomogeneous conormal distribution on an appropriate manifold with corners. Our discussion is a generalization of an earlier work by Albin and Sher, and provides a fundamental first step towards analysis of Ray-Singer torsion, eta-invariants and index theorems in the setting.