Slab Theorem and Halfspace Theorem for constant mean curvature surfaces in $\mathbb H^2\times\mathbb R$

TL;DR

This paper establishes geometric restrictions for constant mean curvature surfaces in hyperbolic space times a line, proving they cannot be contained in certain slabs and must be graphs under specific conditions.

Contribution

It proves new halfspace and slab theorems for constant mean curvature surfaces in cb2b7b2, extending classical results to this setting.

Findings

Properly embedded annular ends with 0<Ha0b1a0b2 cannot be contained in horizontal slabs.

Surfaces with 0<Ha0b1a0b2 in cb2b7b2 and finite topology are graphs.

For H=cb2, the graph is entire.

Abstract

We prove that a properly embedded annular end of a surface in $\mathbb H^2\times\mathbb R$ with constant mean curvature $0<H\leq \frac{1}{2}$ can not be contained in any horizontal slab. Moreover, we show that a properly embedded surface with constant mean curvature $0<H\leq \frac{1}{2}$ contained in $\mathbb H^2\times[0,+\infty)$ and with finite topology is necessarily a graph over a simply connected domain of $\mathbb H^2$. For the case $H=\frac{1}{2}$, the graph is entire.