Heat kernel recurrence on space forms and applications

TL;DR

This paper establishes recurrence relations for heat kernels on hyperbolic spaces and spheres, and applies these to compute heat kernel diagonals and trace asymptotics in odd dimensions.

Contribution

It provides new recurrence relations for heat kernels on space forms and applies them to derive explicit formulas for heat kernel diagonals and asymptotic expansions.

Findings

Recurrence relations for heat kernels on hyperbolic spaces

Recurrence relations for heat kernels on spheres

Explicit formulas for heat kernel diagonals and trace asymptotics in odd dimensions

Abstract

In this paper, we first give a direct proof for two recurrence relations of the heat kernels for hyperbolic spaces in \cite{DM}. Then, by similar computation, we give two similar recurrence relations of the heat kernels for spheres. Finally, as an application, we compute the diagonal of heat kernels for odd dimensional hyperbolic spaces and the heat trace asymptotic expansions for odd dimensional spheres.