Determinant of Laplacian on tori of constant positive curvature with one conical point

TL;DR

This paper derives an explicit formula for the zeta-regularized determinant of the Laplacian on a genus one Riemann surface with a specific conical singularity, advancing spectral geometry understanding.

Contribution

It provides the first explicit expression for the Laplacian determinant on tori with a positive curvature metric and a single conical point, filling a gap in spectral geometry literature.

Findings

Explicit formula for the Laplacian determinant derived

Results applicable to genus one Riemann surfaces with conical singularities

Enhances understanding of spectral invariants on singular surfaces

Abstract

We find an explicit expression for the zeta-regularized determinant of (the Friedrichs extension) of the Laplacian on a compact Riemann surface of genus one with conformal metric of curvature $1$ having a single conical singularity of angle $4\pi$.