Conformal upper bounds for the eigenvalues of the Laplacian and Steklov problem

Hassannezhad, Asma

doi:10.1016/j.jfa.2011.08.003

Cited by 71 publications

(130 citation statements)

References 12 publications

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“…To prove Corollary 5, we recall the following Weyl-type estimate due to P. Buser [2]. (See also [13] and [8].) Theorem 12 ( [2]).…”

Section: Proofs Of Main Resultsmentioning

confidence: 99%

Comparison of Steklov eigenvalues on a domain and Laplacian eigenvalues on its boundary in Riemannian manifolds

Xiong

2018

Journal of Functional Analysis

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We prove that in Riemannian manifolds the k-th Steklov eigenvalue on a domain and the square root of the k-th Laplacian eigenvalue on its boundary can be mutually controlled in terms of the maximum principal curvature of the boundary under sectional curvature conditions. As an application, we derive a Weyl-type upper bound for Steklov eigenvalues. A Pohozaev-type identity for harmonic functions on the domain and the min-max variational characterization of both eigenvalues are important ingredients.

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“…To prove Corollary 5, we recall the following Weyl-type estimate due to P. Buser [2]. (See also [13] and [8].) Theorem 12 ( [2]).…”

Section: Proofs Of Main Resultsmentioning

confidence: 99%

Comparison of Steklov eigenvalues on a domain and Laplacian eigenvalues on its boundary in Riemannian manifolds

Xiong

2018

Journal of Functional Analysis

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“…For other interesting applications of this method, see [22] and [19]. The idea is to construct a family of disjointly supported functions with controlled Rayleigh quotient…”

Section: Methods Of Proofmentioning

confidence: 99%

Isoperimetric control of the Steklov spectrum

Colbois

Soufi

Girouard

2011

Journal of Functional Analysis

112

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We prove that the normalized Steklov eigenvalues of a bounded domain in a complete Riemannian manifold are bounded above in terms of the inverse of the isoperimetric ratio of the domain. Consequently, the normalized Steklov eigenvalues of a bounded domain in Euclidean space, hyperbolic space or a standard hemisphere are uniformly bounded above. On a compact surface with boundary, we obtain uniform bounds for the normalized Steklov eigenvalues in terms of the genus. We also establish a relationship between the Steklov eigenvalues of a domain and the eigenvalues of the Laplace-Beltrami operator on its boundary hypersurface.

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“…In [20] Hassannezhad, combining methods of [6] and [10], obtained upper bounds for eigenvalues of the Laplacian in terms of the conformal invariant MCV (see Definition 1.2) and the volume of the manifold.…”

Section: Previous Workmentioning

confidence: 99%

“…This invariant was recently introduced in a work of Hassannezhad [20]. The conformal invariant MCV is somewhat reminiscent of (but different from) the conformal volume studied by Li and Yau in [22].…”

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confidence: 99%

Width, Ricci curvature, and minimal hypersurfaces

Glynn-Adey¹,

Liokumovich²

2017

J. Differential Geom.

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Abstract. Let (M, g0) be a closed Riemannian manifold of dimension n, for 3 ≤ n ≤ 7, and non-negative Ricci curvature. Let g = φ 2 g0 be a metric in the conformal class of g0. We show that there exists a smooth closed embedded minimal hypersurface in (M, g) of volume bounded by C(n)V n−1 n , where V is the total volume of (M, g). When Ric(M, g0) ≥ −(n − 1) we obtain a similar bound with constant C depending only on n and the volume of (M, g0).Our second result concerns manifolds (M, g) of positive Ricci curvature and dimension at most seven. We obtain an effective version of a theorem of F. C. Marques and A. Neves on the existence of infinitely many minimal hypersurfaces on (M, g) . We show that for any such manifold there exists k minimal hypersurfaces of volume at most CnV sys n−1 (M ), where V denotes the volume of (M, g0) and sys n−1 (M ) is the smallest volume of a non-trivial minimal hypersurface.

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Conformal upper bounds for the eigenvalues of the Laplacian and Steklov problem

Cited by 71 publications

References 12 publications

Comparison of Steklov eigenvalues on a domain and Laplacian eigenvalues on its boundary in Riemannian manifolds

Comparison of Steklov eigenvalues on a domain and Laplacian eigenvalues on its boundary in Riemannian manifolds

Isoperimetric control of the Steklov spectrum

Width, Ricci curvature, and minimal hypersurfaces

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