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

Abstract

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 $H^2\times \mathbb{R}$, we give an upper bound on the infimum of the spectrum of the Laplacian and on the volume growth.

References in corpus (2)

• Minimal hypersurfaces in $\HH^n \times \R$, total curvature and index
• Infimum of the exponential volume growth and the bottom of the essential spectrum of the Laplacian