Eigenvalue estimates for hypersurfaces in $H^m \times R$ and applications

Pierre Bérard (1) , Philippe Castillon (2) , Marcos P. Cavalcante (3)

1 IF - Institut Fourier
2 I3M - Institut de Mathématiques et de Modélisation de Montpellier
3 Instituto de Matematica - UFAL

In this paper, we give a lower bound for the spectrum of the Laplacian on minimal hypersurfaces immersed into $H^m \times R$. As an application, in dimension 2, we prove that a complete minimal surface with finite total extrinsic curvature has finite index. On the other hand, for stable, minimal surfaces in $H^3$ or in $H^2 \times R$, we give an upper bound on the infimum of the spectrum of the Laplacian and on the volume growth.

Mots clés: Minimal hypersurfaces; eigenvalue estimates; stability; index

Domaines: Géométrie différentielle [math.DG]

Pierre Bérard, Philippe Castillon, Marcos P. Cavalcante. Eigenvalue estimates for hypersurfaces in $H^m \times R$ and applications. Pacific Journal of Mathematics, 2011, 253 (1), pp.19-35. ⟨10.2140/pjm.2011.253.19⟩. ⟨hal-00499253⟩