Polyakov-Alvarez Formula for Curvilinear Polygonal Domains with Slits

TL;DR

This paper derives a short time asymptotic expansion of the heat trace and defines the zeta-regularized determinant of the Laplacian on curvilinear polygonal domains with corners and slits, including a comparison formula for conformal metric changes.

Contribution

It extends the Polyakov-Alvarez formula to curvilinear polygonal domains with arbitrary positive angles and slits, providing new tools for spectral analysis on such complex geometries.

Findings

Derived short time asymptotic expansion of the heat trace.

Defined the zeta-regularized determinant for these domains.

Proved a comparison formula for conformal metric changes.

Abstract

We consider the $\zeta$-regularized determinant of the Friedrichs extension of the Dirichlet Laplace-Beltrami operator on curvilinear polygonal domains with corners of arbitrary positive angles. In particular, this includes slit domains. We obtain a short time asymptotic expansion of the heat trace using a classical patchwork method. This allows us to define the $\zeta$-regularized determinant of the Laplacian and prove a comparison formula of Polyakov-Alvarez type for a smooth and conformal change of metric.