Hypersurfaces with H_{r+1}=0 in H x R

TL;DR

This paper constructs and classifies rotational hypersurfaces in hyperbolic space times real line with zero (r+1)-mean curvature, and establishes uniqueness results for r-minimal hypersurfaces with specified boundaries.

Contribution

It introduces new rotational hypersurfaces with specific curvature properties and proves their classification and uniqueness, extending understanding of r-minimal hypersurfaces in hyperbolic products.

Findings

Existence of rotational hypersurfaces with H_{r+1}=0 in H^n x R

Classification of these hypersurfaces

Uniqueness theorems for r-minimal hypersurfaces with boundary

Abstract

We prove the existence of rotational hypersurfaces in $\mathbb{H}^n\times \mathbb{R}$ with $H_{r+1}=0$ and we classify them. Then we prove some uniqueness theorems for $r$-minimal hypersurfaces with a given (finite or asymptotic) boundary. In particular, we obtain a Schoen-type Theorem for two ended complete hypersurfaces.