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

Author:

Jesús Castro-Infantes, José S. Santiago

Keyword:

Mathematics, Differential Geometry, Differential Geometry (math.DG)

journal:

date:

2023-06-29 16:00:00

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.

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