Prescription du spectre de Steklov dans une classe conforme

Jammes, Pierre

doi:10.2140/apde.2014.7.529

Cited by 12 publications

(10 citation statements)

References 35 publications

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“…Remark. While the present paper was at the final stage of preparation, a different proof of inequality (1.3.2) for orientable surfaces (and, consequently, of part (i) of Corollary 1.4.1) appeared in [12,22]. The approaches behind all the proofs go back to the ideas of Cheng and Besson.…”

Section: Resultsmentioning

confidence: 93%

Multiplicity bounds for Steklov eigenvalues on Riemannian surfaces

Karpukhin¹,

Kokarev²,

Polterovich³

2014

Ann. inst. Fourier

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We prove two explicit bounds for the multiplicities of Steklov eigenvalues σ k on compact surfaces with boundary. One of the bounds depends only on the genus of a surface and the index k of an eigenvalue, while the other depends as well on the number of boundary components. We also show that on any given Riemannian surface with smooth boundary the multiplicities of Steklov eigenvalues σ k are uniformly bounded in k.

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Section: Resultsmentioning

confidence: 93%

Multiplicity bounds for Steklov eigenvalues on Riemannian surfaces

Karpukhin¹,

Kokarev²,

Polterovich³

2014

Ann. inst. Fourier

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“…The restrictions of the corresponding eigenfunctions to the boundary form an orthogonal basis in L 2 (M). Geometric properties of Steklov eigenvalues on Riemannian manifolds have been actively investigated in the recent years, see [CEG,FS1,FS2,GP2,Ja,KKP]. In particular, various estimates on eigenvalues and their multiplicities have been obtained.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Heat Invariants of the Steklov Problem

Polterovich

Sher

2013

J Geom Anal

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Abstract. We study the heat trace asymptotics associated with the Steklov eigenvalue problem on a Riemannian manifold with boundary. In particular, we describe the structure of the Steklov heat invariants and compute the first few of them explicitly in terms of the scalar and mean curvatures. This is done by applying the Seeley calculus to the Dirichlet-to-Neumann operator, whose spectrum coincides with the Steklov eigenvalues. As an application, it is proved that a three-dimensional ball is uniquely defined by its Steklov spectrum among all Euclidean domains with smooth connected boundary.

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“…This is in contrast to the case of Laplace eigenvalues on closed surfaces, where there can be points where the density vanishes, and at these point the maximizing metric has conical singularities (though the angles at the conical points are integer multiples of 2π ; see [21], p. [18][19]. In the Steklov case, we have that the maximizing metric g is smooth on the boundary, and can be taken smooth in the interior.…”

Section: Theorem 58mentioning

confidence: 88%

“…These results appeared almost simultaneously in two other papers [18,20], with some improvements in [20]. The approaches behind all the proofs go back to the ideas of Cheng and Besson on multiplicity bounds of the eigenvalues of the Laplacian on closed surfaces.…”

Section: Notation and Preliminariesmentioning

confidence: 89%

Sharp eigenvalue bounds and minimal surfaces in the ball

2015

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We prove existence and regularity of metrics on a surface with boundary which maximize sigma_1 L where sigma_1 is the first nonzero Steklov eigenvalue and L the boundary length. We show that such metrics arise as the induced metrics on free boundary minimal surfaces in the unit ball B^n for some n. In the case of the annulus we prove that the unique solution to this problem is the induced metric on the critical catenoid, the unique free boundary surface of revolution in B^3. We also show that the unique solution on the Mobius band is achieved by an explicit S^1 invariant embedding in B^4 as a free boundary surface, the critical Mobius band. For oriented surfaces of genus 0 with arbitrarily many boundary components we prove the existence of maximizers which are given by minimal embeddings in B^3. We characterize the limit as the number of boundary components tends to infinity to give the asymptotically sharp upper bound of 4pi. We also prove multiplicity bounds on sigma_1 in terms of the topology, and we give a lower bound on the Morse index for the area functional for free boundary surfaces in the ball.Comment: 52 pages. Final version that appeared in Invent. Math.: Presentation of Proposition 4.3 improved, other minor edits throughou

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Prescription du spectre de Steklov dans une classe conforme

Cited by 12 publications

References 35 publications

Multiplicity bounds for Steklov eigenvalues on Riemannian surfaces

Multiplicity bounds for Steklov eigenvalues on Riemannian surfaces

Heat Invariants of the Steklov Problem

Sharp eigenvalue bounds and minimal surfaces in the ball

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