Small time heat kernel asymptotics at the cut locus on surfaces of revolution

TL;DR

This paper studies the small time heat kernel behavior at the cut locus on certain surfaces of revolution, revealing how the exponential map degenerates and affecting the heat kernel asymptotics.

Contribution

It provides the first example where the minimal degeneration of the heat kernel asymptotics at the cut locus is explicitly characterized on surfaces of revolution.

Findings

Degeneracy of the exponential map near a cut-conjugate point is determined.

Consequences of this degeneracy on heat kernel asymptotics are analyzed.

First example of minimal degeneration in heat kernel asymptotics at the cut locus.

Abstract

In this paper we investigate the small time heat kernel asymptotics on the cut locus on a class of surfaces of revolution, which are the simplest 2-dimensional Riemannian manifolds different from the sphere with non trivial cut-conjugate locus. We determine the degeneracy of the exponential map near a cut-conjugate point and present the consequences of this result to the small time heat kernel asymptotics at this point. These results give a first example where the minimal degeneration of the asymptotic expansion at the cut locus is attained.