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Minoration du spectre des variétés hyperboliques de dimension 3

Pierre Jammes (2012)

Bulletin de la Société Mathématique de France

Soit M une variété hyperbolique compacte de dimension 3, de diamètre  d et de volume ≤ V . Si on note μ i ( M ) la i -ième valeur propre du laplacien de Hodge-de Rham agissant sur les 1-formes coexactes de M , on montre que μ 1 ( M ) ≥ c d 3 e 2 k d et μ k + 1 ( M ) ≥ c d 2 , où c > 0 est une constante ne dépendant que de V , et k est le nombre de composantes connexes de la partie mince de M . En outre, on montre que pour toute 3-variété hyperbolique M ∞ de volume fini avec cusps, il existe une suite M i de remplissages compacts de M ∞ , de diamètre d i → + ∞ telle que et μ 1 ( M i ) ≥ c d i 2 .

Multiplicity bounds for Steklov eigenvalues on Riemannian surfaces

Mikhail Karpukhin, Gerasim Kokarev, Iosif Polterovich (2014)

Annales de l’institut Fourier

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 .

Normal form of the wave group and inverse spectral theory

Steve Zelditch (1998)

Journées équations aux dérivées partielles

This talk will describe some results on the inverse spectral problem on a compact riemannian manifold (possibly with boundary) which are based on V. Guillemin's strategy of normal forms. It consists of three steps : first, put the wave group into a normal form around each closed geodesic. Second, determine the normal form from the spectrum of the laplacian. Third, determine the metric from the normal form. We will try to explain all three steps and to illustrate with simple examples such as surfaces...

On 2 ⁢ p -dimensional Riemannian manifolds with positive scalar curvature

Domenico Perrone (1984)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

In questo lavoro si danno alcuni risultati sugli spettri degli operatori di Laplace per varietà Riemanniane compatte con curvatura scalare positiva e di dimensione 2 ⁢ p . Ad essi si aggiunge una osservazione riguardante la congettura di Yamabe.

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