Vol. 253, No. 1, 2011

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Eigenvalue estimates for hypersurfaces in m × and applications

Pierre Bérard, Philippe Castillon and Marcos Cavalcante

Vol. 253 (2011), No. 1, 19–35
Abstract

In the first part of this paper, we give a lower bound for the spectrum of the Laplacian on minimal hypersurfaces immersed into m × . As an application, in dimension 2, we prove that a complete minimal surface with finite total extrinsic curvature has finite index. In the second part, we consider the operator L = Δg + a + bKg on a complete noncompact surface (M2,g). Assuming that L is nonnegative for some constants a > 0 and b > 14, we show that the infimum of the spectrum of M2 is bounded from above by a∕(4b 1). We apply this result to stable minimal surfaces immersed into homogeneous 3-manifolds.

Keywords
minimal hypersurface, eigenvalue estimate, stability, index
Mathematical Subject Classification 2000
Primary: 53C42, 58C40
Milestones
Received: 23 July 2010
Revised: 9 March 2011
Accepted: 11 April 2011
Published: 28 November 2011
Authors
Pierre Bérard
Université Grenoble 1
Institut Fourier (UJF-CNRS), BP 74
38402 Saint Martin d’Hères Cedex
France
Philippe Castillon
Département des sciences mathématiques CC 51
Université Montpellier II,  I3M (UMR 5149)
34095 Montpellier Cedex 5
France
Marcos Cavalcante
Instituto de Matemática
Universidade Federal de Alagoas
57072-900 Maceió AL
Brazil