Compactness of relatively isospectral sets of surfaces via conformal surgeries

TL;DR

This paper introduces a new notion of relative isospectrality for surfaces with boundary and non-compact ends, establishing a compactness result through conformal surgeries that connect to known cases.

Contribution

It defines relative isospectrality for complex surfaces and proves a compactness theorem using conformal surgeries, extending previous results to broader classes of surfaces.

Findings

Established a compactness result for families of relatively isospectral surfaces

Reduced the problem to known cases of hyperbolic surfaces and surfaces with cusps

Extended the understanding of spectral geometry for non-compact surfaces

Abstract

We introduce a notion of relative isospectrality for surfaces with boundary having possibly non-compact ends either conformally compact or asymptotic to cusps. We obtain a compactness result for such families via a conformal surgery that allows us to reduce to the case of surfaces hyperbolic near infinity recently studied by Borthwick and Perry, or to the closed case by Osgood, Phillips and Sarnak if there are only cusps.