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Pinching of the first eigenvalue of the Laplacian for hypersurfaces and rigidity. (Pincement de la première valeur propre du Laplacien pour les hypersurfaces et rigidité.) (French. English summary) Zbl 1187.58010

Actes de Séminaire de Théorie Spectrale et Géométrie. Année 2007–2008. St. Martin d’Hères: Université de Grenoble I, Institut Fourier. Séminaire de Théorie Spectrale et Géométrie 26, 123-138 (2008).
From the author’s abstract: Robert C. Reilly obtained upper bounds for the first eigenvalue of the Laplacian for hypersurfaces of Euclidian 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.
For the entire collection see [Zbl 1173.35005].
Reviewer: Salah Mehdi (Metz)

MSC:

58C40 Spectral theory; eigenvalue problems on manifolds
53A07 Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces
53C20 Global Riemannian geometry, including pinching
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
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