DtN isospectrality, flat metrics with non-trivial holonomy and comparison formula for determinants of Laplacian

TL;DR

This paper investigates the determinants of Laplacians on flat conical surfaces with different holonomy types, providing comparison formulas that highlight the differences between trivial and non-trivial holonomy cases.

Contribution

It introduces new comparison formulas for zeta-regularized determinants of Laplacians on flat conical surfaces, emphasizing the impact of holonomy on these determinants.

Findings

Comparison formulas differ significantly between trivial and non-trivial holonomy cases.

The study advances understanding of spectral invariants on flat conical surfaces.

Results apply to surfaces of genus g ≥ 2.

Abstract

We study comparison formulas for $\zeta$-regularized determinants of self-adjoint extensions of the Laplacian on flat conical surfaces of genus $g\geq 2$. The cases of trivial and non-trivial holonomy of the metric turn out to differ significantly.