Eigenvalue estimates for hypersurfaces in ℍ^m × ℝ and applications

Pierre Bérard, Philippe Castillon and Marcos Cavalcante

Vol. 253 (2011), No. 1, 19–35

DOI: 10.2140/pjm.2011.253.19

Abstract

In the first part of this paper, we give a lower bound for the spectrum of the Laplacian on minimal hypersurfaces immersed into ℍ^m × ℝ. 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 = Δ_g + a + bK_g on a complete noncompact surface (M²,g). Assuming that L is nonnegative for some constants a > 0 and b > 1∕4, 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