Variational formulas for determinant of Laplacian on higher genus polyhedral surface

TL;DR

This paper derives variational formulas for the determinant of the Laplacian on higher genus polyhedral surfaces with conical metrics, providing explicit expressions up to moduli-dependent factors.

Contribution

It introduces new variational formulas for the Laplacian determinant on polyhedral surfaces and expresses it explicitly in terms of moduli and conical data.

Findings

Derived variational formulas for the Laplacian determinant.

Explicit expression for the determinant up to moduli factors.

Potential for calculating the determinant via comparison with known cases.

Abstract

Let $X$ be a Riemann surface of genus $g\ge 1$ endowed with a flat conical metric $m$ and let ${\rm det}\,\Delta$ be the $\zeta$-regularized determinant of the Friedrichs Laplacian on $(X,m)$. We derive variational formulas for ${\rm det}\,\Delta$ with respect to conical points and conical angles within a given conformal class. Integration of them leads to an explicit expression for ${\rm det}\,\Delta$ up to moduli dependent factor. The latter, in principle, can be calculated via comparison of the above result with the well-known formulas for the case of flat conical metrics with trivial holonomy.