Cylindrically bounded constant mean curvature surfaces in $\mathbb{H}^2\times\mathbb{R}$

TL;DR

This paper proves that properly embedded constant mean curvature surfaces in hyperbolic space times the real line, which are finite topology and bounded distance from a vertical geodesic, are rotationally symmetric.

Contribution

It establishes a symmetry result for certain constant mean curvature surfaces in  imes \u211d, extending understanding of their geometric properties.

Findings

Surfaces with finite topology and bounded distance from a vertical geodesic are rotationally symmetric.

The result applies to properly embedded constant mean curvature surfaces in  imes .

Provides a classification of such surfaces based on symmetry properties.

Abstract

In this paper we prove that a properly embedded constant mean curvature surface in $\mathbb{H}^2\times\mathbb{R}$ which has finite topology and stays at a finite distance from a vertical geodesic line is invariant by rotation around a vertical geodesic line.