A pinching theorem for the first eigenvalue of the Laplacian on hypersurfaces of the Euclidean space

  • Bruno Colbois

    Université de Neuchâtel, Switzerland
  • Jean-Francois Grosjean

    Université Henri Poincaré, Vandoeuvre Les Nancy, France

Abstract

In this paper, we give pinching theorems for the first nonzero eigenvalue of the Laplacian on the compact hypersurfaces of the Euclidean space. Indeed, we prove that if the volume of is then, for any , there exists a constant depending on the dimension of and the -norm of the mean curvature , so that if the -norm () of satisfies , then the Hausdorff-distance between and a round sphere of radius is smaller than . Furthermore, we prove that if is a small enough constant depending on and the -norm of the second fundamental form, then the pinching condition implies that is diffeomorphic to an -dimensional sphere.

Cite this article

Bruno Colbois, Jean-Francois Grosjean, A pinching theorem for the first eigenvalue of the Laplacian on hypersurfaces of the Euclidean space. Comment. Math. Helv. 82 (2007), no. 1, pp. 175–195

DOI 10.4171/CMH/88