On Determinants of Laplacians on Compact Riemann Surfaces Equipped with Pullbacks of Conical Metrics by Meromorphic Functions

TL;DR

This paper derives an explicit formula for the zeta-regularized determinant of the Laplace-Beltrami operator on Riemann surfaces with conical metrics pulled back via meromorphic functions, linking geometry, analysis, and moduli space parameters.

Contribution

It provides a novel explicit formula for the Laplacian determinant on Riemann surfaces with conical metrics obtained through pullbacks by meromorphic functions, expanding understanding of spectral invariants.

Findings

Explicit formula for the Laplacian determinant on conical Riemann surfaces.

Analysis of conical singularities induced by pullback metrics.

Connection between spectral invariants and moduli space parameters.

Abstract

Let $\mathsf m$ be any conical (or smooth) metric of finite volume on the Riemann sphere $\Bbb CP^1$. On a compact Riemann surface $X$ of genus $g$ consider a meromorphic funciton $f: X\to {\Bbb C}P^1$ such that all poles and critical points of $f$ are simple and no critical value of $f$ coincides with a conical singularity of $\mathsf m$ or $\{\infty\}$. The pullback $f^*\mathsf m$ of $\mathsf m$ under $f$ has conical singularities of angles $4\pi$ at the critical points of $f$ and other conical singularities that are the preimages of those of $\mathsf m$. We study the $\zeta$-regularized determinant $\operatorname{Det}' \Delta_F$ of the (Friedrichs extension of) Laplace-Beltrami operator on $(X,f^*\mathsf m)$ as a functional on the moduli space of pairs $(X, f)$ and obtain an explicit formula for $\operatorname{Det}' \Delta_F$.