On the structure of hypersurfaces in $\mathbb{H}^n\times \mathbb{R}$ with finite strong total curvature

TL;DR

This paper studies the geometric structure of hypersurfaces in hyperbolic space cross a line with finite strong total curvature, proving properness, topological classification, and analyzing special curvature conditions and symmetries.

Contribution

It establishes properness and topological classification of hypersurfaces with finite strong total curvature in rac{H}^n imes \u007R, and provides new classification and maximum principle results under additional curvature assumptions.

Findings

Hypersurfaces with finite strong total curvature are proper and topologically finite punctured manifolds.

Classification of hypersurfaces with zero higher order mean curvature invariant under hyperbolic translations.

Maximum principle for these hypersurfaces in a half-space setting.

Abstract

We prove that if $X:M^n\to\mathbb{H}^n\times \mathbb{R}$, $n\geq 3$, is a an orientable, complete immersion with finite strong total curvature, then $X$ is proper and $M$ is diffeomorphic to a compact manifold $\bar M$ minus a finite number of points $q_1, \dots q_k$. Adding some extra hypothesis, including $H_r=0,$ where $H_r$ is a higher order mean curvature, we obtain more information about the geometry of a neighbourhood of each puncture. The reader will also find in this paper a classification result for the hypersurfaces of $\mathbb{H}^n\times \mathbb{R}$ which satisfy $H_r=0$ and are invariant by hyperbolic translations and a maximum principle in a half space for these hypersurfaces.