Determinants of Laplacians for constant curvature metrics with three conical singularities on 2-sphere

TL;DR

This paper derives an explicit formula for the spectral determinant of the Laplacian on a 2-sphere with three conical singularities under constant curvature metrics, revealing conditions for extremal and minimal configurations.

Contribution

It provides a closed-form expression for the spectral determinant on singular metrics and identifies stationary and minimal points based on conical singularity parameters.

Findings

Explicit formula for spectral determinant of Laplacian with conical singularities.

Stationary point occurs when conical angles are equal for fixed area and singularity sum.

Minimum of the determinant is achieved when the surface area is sufficiently small and angles are equal.

Abstract

We deduce an explicit closed formula for the zeta-regularized spectral determinant of the Friedrichs Laplacian on the Riemann sphere equipped with arbitrary constant curvature (flat, spherical, or hyperbolic) metric having three conical singularities of order $\beta_j\in(-1,0)$ (or, equivalently, of angle $2\pi(\beta_j+1)$). We show that among the metrics with a fixed value of the sum $\beta_1+\beta_2+\beta_3$ and a fixed surface area, those with $\beta_1=\beta_2=\beta_3$ correspond to a stationary point of the determinant. If, in addition, the surface area is sufficiently small, then the stationary point is a minimum. As a crucial step towards obtaining these results we find a relation between the determinant of Laplacian and the Liouville action introduced by A. Zamolodchikov and Al. Zamolodchikov in connection with the celebrated DOZZ formula for the three-point structure constants of the Liouville field theory.