Polyakov-Alvarez type comparison formulas for determinants of Laplacians on Riemann surfaces with conical singularities

TL;DR

This paper derives comparison formulas for determinants of Laplacians on Riemann surfaces with conical singularities, revealing their dependence on singularity orders and providing explicit formulas and examples.

Contribution

It establishes Polyakov-Alvarez type formulas for determinants of Laplacians with conical singularities and offers rigorous proofs and new explicit formulas for various singular surfaces.

Findings

Determinants depend on conical singularity orders.

Maximum determinant on constant curvature sphere occurs at the standard metric.

Provides rigorous proof of the Aurell-Salomonson formula for polyhedra.

Abstract

We present and prove Polyakov-Alvarez type comparison formulas for the determinants of Friederichs extensions of Laplacians corresponding to conformally equivalent metrics on a compact Riemann surface with conical singularities. In particular, we find how the determinants depend on the orders of conical singularities. We also illustrate these general results with several examples: based on our Polyakov-Alvarez type formulas we recover known and obtain new explicit formulas for determinants of Laplacians on singular surfaces with and without boundary. In one of the examples we show that on the metrics of constant curvature on a sphere with two conical singularities and fixed area $4\pi$ the determinant of Friederichs Laplacian is unbounded from above and attains its local maximum on the metric of standard round sphere. In another example we deduce the famous Aurell-Salomonson formula for the determinant of Friederichs Laplacian on polyhedra with spherical topology, thus providing the formula with mathematically rigorous proof.