Spectral geometry of the Steklov problem on orbifolds

TL;DR

This paper explores how the Steklov spectrum relates to the geometry and topology of Riemannian orbifolds with boundary, revealing its ability to detect boundary singularities and lengths in two dimensions, but not interior singularities.

Contribution

It provides precise asymptotics of the Steklov spectrum on orbifolds, shows spectral detection of boundary features in 2D, and extends eigenvalue bounds to the orbifold setting.

Findings

Steklov spectrum detects boundary singularities and boundary component lengths in 2D.

Examples show the spectrum does not detect interior singularities or orbifold Euler characteristic.

Derived sharp eigenvalue bounds based on orbifold invariants.

Abstract

We consider how the geometry and topology of a compact $n$-dimensional Riemannian orbifold with boundary relates to its Steklov spectrum. In two dimensions, motivated by work of A. Girouard, L. Parnovski, I. Polterovich and D. Sher in the manifold setting, we compute the precise asymptotics of the Steklov spectrum in terms of only boundary data. As a consequence, we prove that the Steklov spectrum detects the presence and number of orbifold singularities on the boundary of an orbisurface and it detects the number each of smooth and singular boundary components. Moreover, we find that the Steklov spectrum also determines the lengths of the boundary components modulo an equivalence relation, and we show by examples that this result is the best possible. We construct various examples of Steklov isospectral Riemannian orbifolds which demonstrate that these two-dimensional results do not extend to higher dimensions. In addition, we give two-dimensional examples which show that the Steklov spectrum does \emph{not} detect the presence of interior singularities nor does it determine the orbifold Euler characteristic. In fact, a flat disk is Steklov isospectral to a cone. In another direction, we obtain upper bounds on the Steklov eigenvalues of a Riemannian orbifold in terms of the isoperimetric ratio and a conformal invariant. We generalize results of B. Colbois, A. El Soufi and A. Girouard, and the fourth author to the orbifold setting; in the process, we gain a sharpness result on these bounds that was not evident in the manifold setting. In dimension two, our eigenvalue bounds are solely in terms of the orbifold Euler characteristic and the number each of smooth and singular boundary components.