Higher dimensional surgery and Steklov eigenvalues

TL;DR

This paper demonstrates that in dimensions three and higher, topological modifications via surgeries minimally affect Steklov eigenvalues, extending previous 1D results and showing that certain domain configurations do not maximize these eigenvalues.

Contribution

It generalizes 1D surgery results to higher dimensions and eigenvalues, revealing topological invariance of Steklov spectra under surgeries and non-maximality of certain domain configurations.

Findings

Surgeries of codimension ≥2 preserve Steklov spectrum approximately

Unit ball does not maximize higher Steklov eigenvalues among contractible domains

Disjoint unions of unit balls do not maximize higher eigenvalues in higher dimensions

Abstract

We show that for compact Riemannian manifolds of dimension at least $3$ with nonempty boundary, we can modify the manifold by performing surgeries of codimension $2$ or higher, while keeping the Steklov spectrum nearly unchanged. This shows that certain changes in the topology of a domain do not have an effect when considering shape optimization questions for Steklov eigenvalues in dimensions $3$ and higher. Our result generalizes the 1-dimensional surgery in [FS2] to higher dimensional surgeries and to higher eigenvalues. It is proved in [FS2] that the unit ball does not maximize the first nonzero normalized Steklov eigenvalue among contractible domains in $\mathbb{R}^n$, for $n \geq 3$. We show that this is also true for higher Steklov eigenvalues. Using similar ideas we show that in $\mathbb{R}^n$, for $n\geq 3$, the $j$-th normalized Steklov eigenvalue is not maximized in the limit by a sequence of contractible domains degenerating to the disjoint union of $j$ unit balls, in contrast to the case in dimension $2$ [GP1].