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Pré-Publication, Document De Travail Année : 2023

Maximal multiplicity of Laplacian eigenvalues in negatively curved surfaces

Multiplicité maximale des valeurs propres du Laplacien pour les surfaces à courbure négative

Cyril Letrouit
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  • PersonId : 1269745
Simon Machado
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  • PersonId : 1269746

Résumé

In this work, we obtain the first upper bound on the multiplicity of Laplacian eigenvalues for negatively curved surfaces which is sublinear in the genus g. Our proof relies on a trace argument for the heat kernel, and on the idea of leveraging an r-net in the surface to control this trace. This last idea was introduced in [Jiang-Tidor-Yao-Zhang-Zhao, 2021] for similar spectral purposes in the context of graphs of bounded degree. Our method is robust enough to also yield an upper bound on the “approximate multiplicity” of eigenvalues, i.e., the number of eigenvalues in windows of size $1/\log^\beta(g)$, $\beta>0$. This work provides new insights on a conjecture by Colin de Verdière [Colin de Verdière, 1986] and new ways to transfer spectral results from graphs to surfaces.
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Dates et versions

hal-04160247 , version 1 (12-07-2023)
hal-04160247 , version 2 (30-04-2024)

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Cyril Letrouit, Simon Machado. Maximal multiplicity of Laplacian eigenvalues in negatively curved surfaces. 2023. ⟨hal-04160247v2⟩
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