Minimal hypersurfaces in $\HH^n \times \R$, total curvature and index

Abstract

In this paper, we consider minimal hypersurfaces in the product space ${\mathbb{H}}^n\times \mathbb{R}$. We begin by studying examples of rotation hypersurfaces and hypersurfaces invariant under hyperbolic translations. We then consider minimal hypersurfaces with finite total curvature. This assumption implies that the corresponding curvature goes to zero uniformly at infinity. We show that surfaces with finite total intrinsic curvature have finite index. The converse statement is not true as shown by our examples which also serve as useful barriers.

Reference fixed and comments added

References in corpus (3)

• Lindelöf's theorem for catenoids revisited
• Stability properties for the higher dimensional catenoid in $\rr^{n+1}$
• Minimal graphs in H^n xR and R^{n+1}

Cited by in corpus (6)

• Lindelöf's theorem for catenoids revisited
• Stability Properties of Rotational Catenoids in the Heisenberg Groups
• Hypersurfaces with H_{r+1}=0 in H x R
• Caccioppoli's inequalities on constant mean curvature hypersurfaces in Riemannian manifolds
• Eigenvalue estimates for hypersurfaces in $H^m\times R$ and applications
• Examples of $H$-hypersurfaces in $H^n\times R$ and geometric applications