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Soit une variété hyperbolique compacte de dimension 3, de diamètre et de volume . Si on note la -ième valeur propre du laplacien de Hodge-de Rham agissant sur les 1-formes coexactes de , on montre que et , où est une constante ne dépendant que de , et est le nombre de composantes connexes de la partie mince de . En outre, on montre que pour toute 3-variété hyperbolique de volume fini avec cusps, il existe une suite de remplissages compacts de , de diamètre telle que et .
We prove two explicit bounds for the multiplicities of Steklov eigenvalues on compact surfaces with boundary. One of the bounds depends only on the genus of a surface and the index 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 are uniformly bounded in .
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...
In questo lavoro si danno alcuni risultati sugli spettri degli operatori di Laplace per varietà Riemanniane compatte con curvatura scalare positiva e di dimensione . Ad essi si aggiunge una osservazione riguardante la congettura di Yamabe.
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