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

Abstract

In this paper we continue with the analysis of spectral problems in the setting of complete manifolds with fibred boundary metrics, also referred to as $$\phi $$ ϕ -metrics, as initiated in our previous work (Grieser et al. in Spectral geometry on manifolds with fibred boundary metrics I: Low energy resolvent, 2020). We consider the Hodge Laplacian for a $$\phi $$ ϕ -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.