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Multiple tubular excisions and large Steklov eigenvalues

Jade Brisson

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 $Ω_\varepsilon\subset M$ 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.

Multiple tubular excisions and large Steklov eigenvalues

Abstract

Given a closed Riemannian manifold and closed connected submanifolds of codimension at least , we prove that the first non-zero eigenvalue of the domain obtained by removing the tubular neighbourhood of size around each tends to infinity as tends to . More precisely, we prove a lower bound in terms of , , the geometry of and the codimensions and the volumes of the submanifolds and an upper bound in terms of and the codimensions of the submanifolds. For eigenvalues of index , we have a stronger result: their order of divergence is and their rate of divergence is only depending on and on the codimensions of the submanifolds.
Paper Structure (8 sections, 10 theorems, 107 equations)

This paper contains 8 sections, 10 theorems, 107 equations.

Table of Contents

  1. Introduction
  2. Tubular excision
  3. Plan of the paper
  4. Notation and convention
  5. Quasi-isometry
  6. Upper bound for the first $b-1$ non-zero eigenvalues and rate of divergence for the truncated spectrum
  7. Large first Steklov eigenvalue for manifolds with multiple tubular excisions
  8. Acknowledgements

Key Result

Theorem 1.2

Let $M$ be a closed Riemannian manifold of dimension $m\geq 2$. Let $N_1\,,\ldots\,,N_b\subset M$ be $b\geq2$ disjoint closed connected submanifolds such that the dimension of $N_j$ is $0\leq n_j\leq m-2$. For $\varepsilon>0$ small enough, the truncated Steklov spectrum of $\Omega_\varepsilon:=M\bac Each family $\mathcal{S}(N_j)$ is the contribution of a submanifold $N_j$ and is given by if $N_j$

Theorems & Definitions (17)

  • Definition 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Example 1.6
  • Definition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Lemma 3.1
  • ...and 7 more