Compact manifolds with fixed boundary and large Steklov eigenvalues

TL;DR

This paper demonstrates how to manipulate Steklov eigenvalues on compact manifolds with boundary through conformal perturbations, achieving arbitrarily large eigenvalues while controlling boundary component proximity, contrasting with Laplace eigenvalue behavior.

Contribution

It introduces methods to increase Steklov eigenvalues via conformal changes supported near the boundary, revealing new spectral properties of manifolds with boundary.

Findings

Arbitrarily large (b+1)-th Steklov eigenvalues can be achieved with boundary-supported conformal perturbations.

Large eigenvalues for j<b+1 require interior spreading of the conformal factor.

Large Steklov eigenvalues can be obtained while keeping boundary components close, using Riemannian submersion.

Abstract

Let M be a compact Riemannian manifold with boundary. Let b>0 be the number of connected components of its boundary. For manifolds of dimension at least 3, we prove that it is possible to obtain an arbitrarily large (b+1)-th Steklov eigenvalue using a smooth conformal perturbation which is supported in a thin neighbourhood of the boundary, identically equal to 1 on the boundary. For j<b+1, it is also possible to obtain arbitrarily large j-th eigenvalue, but this require the conformal factor to spread throughout the interior of the manifold M. This is in stark contrast with the situation for the eigenvalues of the Laplace operator on a closed manifold, where a conformal factor that is large enough for the volume to become unbounded results in the spectrum collapsing to 0. We also prove that it is possible to obtain large Steklov eigenvalues while keeping different boundary components arbitrarily close to each other, by constructing a convenient Riemannian submersion.