Tubular excision and Steklov eigenvalues

TL;DR

This paper investigates how the Steklov eigenvalues behave when a tubular neighborhood around a submanifold is removed from a closed manifold, revealing their divergence rates and implications for spectral gap bounds.

Contribution

It provides a detailed analysis of the asymptotic behavior of Steklov eigenvalues in tubular excisions, highlighting the dependence on codimension and applications to spectral gap and isoperimetric bounds.

Findings

Eigenvalues in mid-frequency range tend to infinity depending on codimension.

High-frequency eigenvalues form unbounded clusters tending to infinity.

Constructs manifolds with unbounded perimeter-normalized spectral gap.

Abstract

Given a closed manifold $M$ and a closed connected submanifold $N\subset M$ of positive codimension, we study the Steklov spectrum of the domain $\Omega_\varepsilon\subset M$ obtained by removing the tubular neighbourhood of size $\varepsilon$ around $N$. All non-zero eigenvalues in the mid-frequency range tend to infinity at a rate which depends only on the codimension of $N$ in $M$. Eigenvalues above the mid-frequency range are also described: they tend to infinity following an unbounded sequence of clusters. This construction is then applied to obtain manifolds with unbounded perimeter-normalized spectral gap and to show the necessity of using the injectivity radius in some known isoperimetric-type upper bounds.