Asymptotics of relative heat traces and determinants on open surfaces of finite area

TL;DR

This paper establishes the existence and asymptotic expansion of the relative heat trace and determinant for Laplace operators on open surfaces with cusp ends, extending spectral geometry results to non-compact settings.

Contribution

It proves the well-definedness of the relative determinant on surfaces with cusps and derives Polyakov's formula in this context, advancing spectral analysis on open surfaces.

Findings

Existence of the relative determinant under conformal conditions

Asymptotic expansion of the relative heat trace for small times

Conditions on the conformal factor for determinant existence

Abstract

The goal of this paper is to prove that on surfaces with asymptotically cusp ends the relative determinant of pairs of Laplace operators is well defined. We consider a surface with cusps (M,g) and a metric h on the surface that is a conformal transformation of the initial metric g. We prove the existence of the relative determinant of the pair $(\Delta_{h},\Delta_{g})$ under suitable conditions on the conformal factor. The core of the paper is the proof of the existence of an asymptotic expansion of the relative heat trace for small times. We find the decay of the conformal factor at infinity for which this asymptotic expansion exists and the relative determinant is defined. Following the paper by B. Osgood, R. Phillips and P. Sarnak about extremal of determinants on compact surfaces, we prove Polyakov's formula for the relative determinant and discuss the extremal problem inside a conformal class. We discuss necessary conditions for the existence of a maximizer.