Vertical Ends of Constant Mean Curvature H=1/2 in H^2\times R

TL;DR

This paper proves a vertical halfspace theorem for properly immersed constant mean curvature surfaces with H=1/2 in the hyperbolic plane cross real line, using geometric maximum principles and rotational surfaces.

Contribution

It establishes a new halfspace theorem for H=1/2 surfaces in H^2×R, extending classical results with a geometric PDE approach.

Findings

Proves a vertical halfspace theorem for H=1/2 surfaces.

Uses maximum principle and rotational surfaces in the proof.

Extends understanding of constant mean curvature surfaces in hyperbolic product spaces.

Abstract

We prove a vertical halfspace theorem for surfaces with constant mean curvature $H={1/2},$ properly immersed in the product space $\h^2\times\re,$ where $\h^2$ is the hyperbolic plane and $\re$ is the set of real numbers. The proof is a geometric application of the classical maximum principle for second order elliptic PDE, using the family of non compact rotational $H=1/2$ surfaces in $\h^2\times\re.$