Examples of $H$-hypersurfaces in $H^n \times R$ and geometric applications

TL;DR

This paper classifies and constructs various $H$-hypersurfaces in $H^n imes R$, providing geometric applications such as barriers, symmetry, and uniqueness results for certain hypersurfaces with prescribed boundary conditions.

Contribution

It describes all rotation and translation $H$-hypersurfaces in $H^n imes R$ and applies them to solve geometric problems involving existence, symmetry, and boundary value conditions.

Findings

Complete classification of rotation $H$-hypersurfaces.

Construction of translation $H$-hypersurfaces generated by convex curves.

Existence of non-entire vertical graphs with infinite boundary data.

Abstract

In this paper we describe all rotation $H$-hypersurfaces in $H^n \times R$ and use them as barriers to prove existence and characterization of certain vertical $H$-graphs and to give symmetry and uniqueness results for compact $H$-hypersurfaces whose boundary is one or two parallel submanifolds in slices. We also describe examples of translation $H$-hypersurfaces in $H^n \times R$. For $n>2$ we obtain a complete embedded translation hypersurface generated by a compact, simple, strictly convex curve. When $0 < H < \frac{n-1}{n}$ we obtain a complete non-entire vertical graph over the non-mean convex domain bounded by an equidistant hypersurface taking infinite boundary value data and infinite asymptotic boundary value data.