The heat kernel on an asymptotically conic manifold

TL;DR

This paper provides a detailed analysis of the heat kernel's long-time behavior on asymptotically conic manifolds, utilizing microlocal analysis to explore spectral invariants like the zeta function and Laplacian determinant.

Contribution

It offers a comprehensive description of the heat kernel's asymptotics on asymptotically conic manifolds and applies this to spectral invariants.

Findings

Complete asymptotic description of the heat kernel in all regimes

Definition and analysis of a renormalized zeta function

Investigation of the Laplacian determinant on asymptotically conic manifolds

Abstract

In this paper, we investigate the long-time structure of the heat kernel on a Riemannian manifold M which is asymptotically conic near infinity. Using geometric microlocal analysis and building on results of Guillarmou and Hassell on the low-energy resolvent, we give a complete description of the asymptotic structure of the heat kernel in all spatial and temporal regimes. We apply this structure to define and investigate a renormalized zeta function and determinant of the Laplacian on M.