Tubular excision and Steklov eigenvalues

2021/07/28 by Jade Brisson, Brisson, Jade · 2 citations
Mathematics · Physics and Astronomy · #35P15 (Secondary) #58J50 (Primary) #FOS: Mathematics #Laser-Matter Interactions and Applications #Quantum chaos and dynamical systems #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2107.13606

openalex publication_date 2021/07/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Given a closed manifold M and a closed connected submanifold N⊂ M of positive codimension, we study the Steklov spectrum of the domain Ω_ε⊂ M obtained by removing the tubular neighbourhood of size ε 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.

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