Genus one $H$-surfaces with $k$-ends in $\mathbb{H}^2\times\mathbb{R}$

TL;DR

This paper constructs new genus one constant mean curvature surfaces with multiple ends in hyperbolic space times a line, demonstrating the absence of a Schoen-type uniqueness theorem for such surfaces.

Contribution

It introduces two families of properly immersed genus one H-surfaces with multiple ends in H^2×R, constructed via conjugate methods, showing non-uniqueness in this setting.

Findings

Surfaces are asymptotic to vertical H-cylinders for 0<H<1/2.

Existence of multiple non-unique genus one H-surfaces with k ends.

No Schoen-type theorem applies to these immersed surfaces.

Abstract

We construct two different families of properly Alexandrov-immersed surfaces in $\mathbb{H}^2\times \mathbb{R}$ with constant mean curvature $0<H\leq \frac 1 2$, genus one and $k\geq2$ ends ($k=2$ only for one of these families). These ends are asymptotic to vertical $H$-cylinders for $0<H<\frac 1 2$. This shows that there is not a Schoen-type theorem for immersed surfaces with positive constant mean curvature in $\mathbb{H}^2\times\mathbb{R}$. These surfaces are obtained by means of a conjugate construction.