Non-properly Embedded H-Planes in H^2xR

TL;DR

This paper constructs complete, stable, simply-connected surfaces with constant mean curvature H in the hyperbolic plane cross real line, which are embedded but not properly embedded, for H in (0, 1/2).

Contribution

It introduces the first examples of non-properly embedded H-planes in H^2×R for H in (0, 1/2), expanding understanding of constant mean curvature surfaces.

Findings

Existence of non-proper, stable H-planes in H^2×R for H in (0, 1/2)

Construction of complete, simply-connected, embedded surfaces with constant mean curvature

Surfaces are non-properly embedded despite stability and completeness

Abstract

For any $H$ in (0,1/2), we construct complete, non-proper, stable, simply-connected surfaces embedded in $H^2xR$ with constant mean curvature $H$.