On the topology of constant mean curvature surfaces in H2 X R with boundary in a plane

TL;DR

This paper investigates the topology of constant mean curvature surfaces in hyperbolic space cross real line, focusing on boundary conditions and their implications for surface topology, extending known results from Euclidean space to hyperbolic settings.

Contribution

It addresses a gap in the proof regarding the topological classification of certain constant mean curvature surfaces in H2 x R, extending classical Euclidean results to hyperbolic space.

Findings

Surfaces with mean curvature close to 1/2 and boundary curvature > 1 are topologically disks under certain conditions.

Identifies a gap in the existing proof that prevents a complete topological classification.

Extends classical results from Euclidean space to hyperbolic space cross R.

Abstract

A gap in the proof prevents us to show that surfaces with constant mean curvature closed to 1/2 in H2 X R and having boundary with curvature greater than one, contained in a horizontal section P of H2 X R are topological disks, provided they are contained in one of the two halfspaces determined by P. This is the analogue in H2 X R of a result in R3 by A. Ros and H. Rosenberg [13, Theorem 2].