Hypersurfaces with Prescribed Boundary and Small Steklov Eigenvalues

Colbois, Bruno; Girouard, Alexandre; Métras, Antoine

doi:10.4153/s000843951900050x

Can. Math. Bull.

2019

DOI: 10.4153/s000843951900050x

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Hypersurfaces with Prescribed Boundary and Small Steklov Eigenvalues

Bruno Colbois¹,

Alexandre Girouard²,

Antoine Métras³

Abstract: Given a smooth compact hypersurface M with boundary Σ = ∂M , we prove the existence of a sequence M j of hypersurfaces with the same boundary as M , such that each Steklov eigenvalue σ k (M j ) tends to zero as j tends to infinity. The hypersurfaces M j are obtained from M by a local perturbation near a point of its boundary. Their volumes and diameters are arbitrarily close to those of M , while the principal curvatures of the boundary remain unchanged.

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Cited by 2 publications

References 19 publications

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Some recent developments on the Steklov eigenvalue problem

Colbois,

Girouard,

Gordon

et al. 2023

Rev Mat Complut

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The Steklov eigenvalue problem, first introduced over 125 years ago, has seen a surge of interest in the past few decades. This article is a tour of some of the recent developments linking the Steklov eigenvalues and eigenfunctions of compact Riemannian manifolds to the geometry of the manifolds. Topics include isoperimetric-type upper and lower bounds on Steklov eigenvalues (first in the case of surfaces and then in higher dimensions), stability and instability of eigenvalues under deformations of the Riemannian metric, optimisation of eigenvalues and connections to free boundary minimal surfaces in balls, inverse problems and isospectrality, discretisation, and the geometry of eigenfunctions. We begin with background material and motivating examples for readers that are new to the subject. Throughout the tour, we frequently compare and contrast the behavior of the Steklov spectrum with that of the Laplace spectrum. We include many open problems in this rapidly expanding area.

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Some recent developments on the Steklov eigenvalue problem

Colbois,

Girouard,

Gordon

et al. 2023

Rev Mat Complut

View full text Add to dashboard Cite

show abstract

The Steklov and Laplacian spectra of Riemannian manifolds with boundary

Colbois

Girouard

Hassannezhad

2020

Journal of Functional Analysis

Self Cite

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Given two compact Riemannian manifolds M 1 and M 2 such that their respective boundaries Σ 1 and Σ 2 admit neighbourhoods Ω 1 and Ω 2 which are isometric, we prove the existence of a constant C such that |σ k (M 1 ) − σ k (M 2 )| ≤ C for each k ∈ N. The constant C depends only on the geometry of Ω 1 ∼ = Ω 2 . This follows from a quantitative relationship between the Steklov eigenvalues σ k of a compact Riemannian manifold M and the eigenvalues λ k of the Laplacian on its boundary. Our main result states that the difference |σ k − √ λ k | is bounded above by a constant which depends on the geometry of M only in a neighbourhood of its boundary. The proofs are based on a Pohozaev identity and on comparison geometry for principal curvatures of parallel hypersurfaces. In several situations, the constant C is given explicitly in terms of bounds on the geometry of Ω 1 ∼ = Ω 2 .1991 Mathematics Subject Classification. 35P15 (primary), 58C40, 35P20 (secondary).1 The notation O(k −∞ ) designates a quantity which tends to zero faster than any power of k.

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Hypersurfaces with Prescribed Boundary and Small Steklov Eigenvalues

Cited by 2 publications

References 19 publications

Some recent developments on the Steklov eigenvalue problem

Some recent developments on the Steklov eigenvalue problem

The Steklov and Laplacian spectra of Riemannian manifolds with boundary

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