Constant mean curvature hypersurfaces in $\mathbb H^n\times\mathbb R$ with small planar boundary

TL;DR

This paper proves that small, pinched boundary constant mean curvature hypersurfaces in hyperbolic space cross a line are topological disks, extending a known Euclidean result to hyperbolic geometry.

Contribution

It establishes a topological disk theorem for constant mean curvature hypersurfaces in hyperbolic space with small boundary contained in a horizontal slice.

Findings

Hypersurfaces are topological disks under given conditions.

Results extend Euclidean theorems to hyperbolic space.

Provides geometric conditions for hypersurface topology.

Abstract

We show that constant mean curvature hypersurfaces in $\mathbb H^n\times\mathbb R$, with small and pinched boundary contained in a horizontal slice $P$ are topological disks, provided they are contained in one of the two halfspaces determined by $P$. This is the analogous in $\mathbb H^n\times\mathbb R$ of a result in $\mathbb R^3$ by A. Ros and H. Rosenberg.