Periodic constant mean curvature surfaces in \(\mathbb{H}^2 \times \mathbb{R}\) (Q488058)

The authors consider periodic surfaces in \(\mathbb{H}^2\times \mathbb{R}\), where \(\mathbb{H}^2\) is the hyperbolic plane defined by the Poincaré disk model. Denoting \(\mathbb{M}=\mathbb{H}^2\times \mathbb{R}/G\), where \(G\) is the discrete group of isometries of \(\mathbb{H}^2\times \mathbb{R}\) generated by horizontal translations along geodesics and/or a vertical translation, they prove an Alexandrov-type theorem for double periodic H-surfaces \(\Sigma \subset \mathbb{M}\) (Theorem 3.1). Various examples of periodic minimal surfaces in \(\mathbb{H}^2\times \mathbb{R}\) are presented.