Valeurs propres du laplacien sur les variétés compactes et géométrie.

Lundi 6 avril 2009 14:00-15:00 - A. el Soufi - Tours

Résumé : The sequence of eigenvalues of the Dirichlet Laplacian on a bounded Euclidean domain satisfies several restrictive conditions such as :
Faber-Krahn isoperimetric inequality, that is the principal eigenvalue is bounded above in terms of the volume of the domain, Payne-Pólya-Weinberger type universal inequalities, that is the $k$-th eigenvalue is controlled in terms of the $k-1$ previous ones, etc.
The situation changes completely as soon as Euclidean domains are replaced by compact manifolds. For example, according to results by Colin de Verdière and Lohkamp, given any compact manifold $M$ of dimension $n\ge 3$, it is possible to prescribe arbitrarily and simultaneously, through the choice of a suitable Riemannian metric on
$M$, a finite part of the spectrum of the Laplacian, the volume and the integral of the scalar curvature. Hence, Faber-Krahn and Payne-Pólya-Weinberger inequalities have no analogue in this context.
In this talk, we will discuss the effect of the geometry on the eigenvalues. We will try to understand what kind of geometric situations lead to large eigenvalues for the Laplacian on manifolds of fixed volume, and what does such a Riemannian manifold look like once realized as a submanifold of a Euclidean space.
On the other hand, we show that when the Laplacian is penalized by the squared norm of the mean curvature, then we obtain a Schrödinger type operator whose spectral behavior is very close to that of the Dirichlet Laplacian on Euclidean domains.

Lieu : bât. 425 - 113-115

Valeurs propres du laplacien sur les variétés compactes et géométrie.  Version PDF