One can hear the corners of a drum

TL;DR

This paper proves that the spectral data of a domain uniquely determines whether it has corners, establishing corners as an elementary geometric spectral invariant in spectral geometry.

Contribution

It demonstrates that corners are spectrally detectable and uniquely determined by the spectrum among domains with Lipschitz boundaries.

Findings

Corners are spectrally determined in simply connected domains.

No isospectrality exists between domains with and without corners.

Corners can be heard as a spectral invariant.

Abstract

We prove that the presence or absence of corners is spectrally determined in the following sense: any simply connected domain with piecewise smooth Lipschitz boundary cannot be isospectral to any connected domain, of any genus, which has smooth boundary. Moreover, we prove that amongst all domains with Lipschitz, piecewise smooth boundary and fixed genus, the presence or absence of corners is uniquely determined by the spectrum. This means that corners are an elementary geometric spectral invariant; one can hear corners.