Small eigenvalues of surfaces: old and new

Ballmann, Werner; Matthiesen, Henrik; Mondal, Sugata

doi:10.4310/iccm.2018.v6.n2.a3

Cited by 5 publications

(2 citation statements)

References 17 publications

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“…Going back to the original claim we observe that if λ c 2g−2 (S m ) 1 4 then each S m has at least 2g − 2 + n non-zero small eigenvalues. This is a contradiction to [29,Theorem 2].…”

Section: Some Existence Resultsmentioning

confidence: 84%

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Topological properties of eigenfunctions of Riemannian surfaces

Mondal¹

2019

Annales De La Faculté Des Sciences De Toulouse : Mathématiques

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“…Going back to the original claim we observe that if λ c 2g−2 (S m ) 1 4 then each S m has at least 2g − 2 + n non-zero small eigenvalues. This is a contradiction to [29,Theorem 2].…”

Section: Some Existence Resultsmentioning

confidence: 84%

“…The details of the above results can, of course, be found in [2] and in [3]. We avoid giving complete details of the arguments here mainly because the author and his collaborators have written a recent survey [1] precisely on this topic.…”

Section: Main Issues In the Extension Theorem 43mentioning

confidence: 99%

Topological properties of eigenfunctions of Riemannian surfaces

Mondal¹

2019

Annales De La Faculté Des Sciences De Toulouse : Mathématiques

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Spectral Instability of Coverings

Ballmann,

Mondal

2024

International Mathematics Research Notices

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Degenerating hyperbolic surfaces and spectral gaps for large genus

Wu,

Zhang,

Zhu

2024

Analysis & PDE

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We study the differences of two consecutive eigenvalues λ i − λ i−1 , i up to 2g − 2, for the Laplacian on hyperbolic surfaces of genus g, and show that the supremum of such spectral gaps over the moduli space has infimum limit at least 1 4 as the genus goes to infinity. A min-max principle for eigenvalues on degenerating hyperbolic surfaces is also established. Theorem 4.1. Let {η(g)} ∞ g=2 be any sequence of integers with η(g) ∈ [1, 2g − 2]. Then lim inf g→∞ sup X g ∈M g SpG η(g) (X g ) ≥ 1 4 .

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Small eigenvalues of surfaces: old and new

Abstract: We discuss our recent work on small eigenvalues of surfaces.As an introduction, we present and extend some of the by now classical work of Buser and Randol and explain novel ideas from articles of Sévennec, Otal, and Otal-Rosas which are of importance in our line of thought.

Cited by 5 publications

References 17 publications

Topological properties of eigenfunctions of Riemannian surfaces

Topological properties of eigenfunctions of Riemannian surfaces

Spectral Instability of Coverings

Degenerating hyperbolic surfaces and spectral gaps for large genus

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