Multiple tubular excisions and large Steklov eigenvalues

TL;DR

This paper investigates how the first non-zero Steklov eigenvalue of a manifold with multiple tubular excisions around submanifolds diverges as the excision radius shrinks, providing bounds and divergence rates.

Contribution

It establishes new bounds and precise divergence rates for eigenvalues on manifolds with multiple tubular excisions, extending previous single-excision results.

Findings

First eigenvalue tends to infinity as excision radius approaches zero.

Derived explicit lower and upper bounds depending on geometry and codimensions.

Eigenvalues of higher index diverge at a rate proportional to 1/ε.

Abstract

Given a closed Riemannian manifold $M$ and $b\geq2$ closed connected submanifolds $N_j\subset M$ of codimension at least $2$, we prove that the first non-zero eigenvalue of the domain $\Omega_\varepsilon\subset M$ obtained by removing the tubular neighbourhood of size $\varepsilon$ around each $N_j$ tends to infinity as $\varepsilon$ tends to $0$. More precisely, we prove a lower bound in terms of $\varepsilon$, $b$, the geometry of $M$ and the codimensions and the volumes of the submanifolds and an upper bound in terms of $\varepsilon$ and the codimensions of the submanifolds. For eigenvalues of index $k=b\,,b+1\,,\ldots$, we have a stronger result: their order of divergence is $\varepsilon^{-1}$ and their rate of divergence is only depending on $m$ and on the codimensions of the submanifolds.