129 research outputs found
The supremum of conformally covariant eigenvalues in a conformal class
Let (M,g) be a compact Riemannian manifold of dimension >2. We show that
there is a metric h conformal to g and of volume 1 such that the first positive
eigenvalue the conformal Laplacian with repect to h is arbitrarily large. A
similar statement is proven for the first positive eigenvalue of the Dirac
operator on a spin manifold of dimension >1
Minoration du spectre des variétés hyperboliques de dimension 3
20 pages, 1 figureInternational audienceLet be a compact hyperbolic 3-manifold of diameter and volume . If denotes the -th egenvalue of the Hodge laplacian acting on coexact 1-forms of , we prove that and , where depends only on , and is the number of connected component of the thin part of . Moreover, we prove that for any finite volume hyperbolic 3-manifold with cusps, there is a sequence of compact fillings of of diameter such that .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
Prescription du spectre de Steklov dans une classe conforme
On any compact manifold of dimension with boundary, we prescibe any
finite part of the Steklov spectrum whithin a given conformal class. In
particular, we prescribe the multiplicity of the first eigenvalues. On a
compact surface with boundary, we show that the multiplicity of the -th
eigenvalue is bounded independently of the metric. On the disk, we give more
precise results : the multiplicity of the first and second positive eigenvalues
are at most 2 and 3 respectively. For the Steklov-Neumann problem on the disk,
we prove that the multiplicity of the -th positive eigenvalue is at most
.Comment: 27pages, in French, 1 figur
Prescription de la multiplicit\'e des valeurs propres du laplacien de Hodge-de Rham
On any compact manifold of dimension greater than 6, we prescribe the volume
and any finite part of the spectrum Hodge Laplacian acting on -form for
. In particular, we prescribe the multiplicity of the first
eigenvalues.Comment: 19 pages, in frenc
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