Triangulations of singular constant curvature spheres via Belyi functions and determinants of Laplacians

TL;DR

This paper derives explicit formulas for the spectral determinants of Laplacians on singular constant curvature spheres constructed via triangulations, linking them to Belyi functions and classical geometric structures.

Contribution

It provides explicit expressions for spectral determinants on singular spheres formed by triangulations, connecting them to Belyi functions and known Laplacian determinants.

Findings

Explicit formulas for spectral determinants on triangulated spheres.

Determinants expressed in terms of Belyi functions and known Laplacian determinants.

Identification of stationary points corresponding to Platonic surfaces.

Abstract

We study the zeta-regularized spectral determinant of the Friedrichs Laplacians on the singular spheres obtained by cutting and glueing copies of constant curvature (hyperbolic, spherical, or flat) double triangle. The determinant is explicitly expressed in terms of the corresponding Belyi functions and the determinant of the Friedrichs Laplacian on the double triangle. The latter determinant was found in a closed explicit form in [V. Kalvin, Calc. Var. 62 (2023), Paper 59, arXiv:2112.02771]. In examples we consider the cyclic, dihedral, tetrahedral, octahedral, and icosahedral triangulations, and find the determinant for the corresponding spherical, Euclidean, and hyperbolic Platonic surfaces. These surfaces correspond to stationary points of the determinant.