Large spectral gaps for Steklov eigenvalues under volume constraints and under localized conformal deformations

TL;DR

This paper demonstrates that it is possible to achieve arbitrarily large Steklov spectral gaps on compact manifolds with fixed boundary by constructing specific metrics and localized conformal deformations, revealing new spectral geometric phenomena.

Contribution

The authors construct manifolds with fixed boundary and unit volume that admit metrics with arbitrarily large Steklov spectral gaps, and analyze the impact of localized conformal deformations on these gaps.

Findings

Spectral gap can be made arbitrarily large with fixed boundary geometry.

Localized conformal deformations can significantly increase the spectral gap.

Support of deformations must connect boundary components to affect the spectral gap.

Abstract

In this paper we construct compact manifolds with fixed boundary geometry which admit Riemannian metrics of unit volume with arbitrarily large Steklov spectral gap. We also study the effect of localized conformal deformations that fix the boundary geometry. For instance, we prove that it is possible to make the spectral gap arbitrarily large using conformal deformations which are localized on domains of small measure, as long as the support of the deformations contains and connects each component of the boundary.