Pincement de la première valeur propre du laplacien pour les hypersurfaces et rigidité

Roth, Julien

Séminaire de théorie spectrale et géométrie, Tome 26 (2007-2008), pp. 123-138.

Robert C. Reilly a obtenu des majorations de la première valeur propre du laplacien pour les hypersurfaces de l’espace euclidien. De plus, il a montré que le cas d’égalité dans ces majorations est atteint uniquement pour les sphères géodésiques. Dans cet exposé, nous nous intéressons au problème de pincement pour ces majorations. Nous montrons que si le cas d’égalité est presque atteint, alors l’hypersurface est proche d’une sphère, en un sens que nous préciserons. Nous déduisons ensuite des résultats pour les hypersurfaces presque ombiliques ainsi qu’une nouvelle caractérisation des sphères géodésiques.

Robert C. Reilly obtained upper bounds for the first eigenvalue of the Laplacian for hypersurfaces of Euclidean space. He also showed that the equality case of these upper bounds is attained only for geodesic spheres. In this talk, we are interested in the pinching problem for these inequalities. We show that if the equality is almost attained, then the hypersurface is close to a sphere. Then, we deduce results for almost umbilical hypersurfaces and a new characterization of geodesic spheres.

Classification : 53A07, 53C20, 53C21, 58C40
Mot clés : hypersurfaces, laplacien, valeurs propres, pincement, rigidité, courbure moyenne constante
Mots clés : Hypersurfaces, Laplacian, Eigenvalues, Pinching, Rigidity, Constant Mean Curvature

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