The Alexandrov problem in a quotient space of ${\mathbb{H}}^2\times \mathbb{R}$

Abstract

We prove an Alexandrov type theorem for a quotient space of ${\mathbb{H}}^2\times \mathbb{R}$. More precisely we classify the compact embedded surfaces with constant mean curvature in the quotient of ${\mathbb{H}}^2\times \mathbb{R}$ by a subgroup of isometries generated by a parabolic translation along horocycles of ${\mathbb{H}}^2$ and a vertical translation. Moreover, we construct some examples of periodic minimal surfaces in ${\mathbb{H}}^2\times \mathbb{R}$ and we prove a multi-valued Rado theorem for small perturbations of the helicoid in ${\mathbb{H}}^2\times \mathbb{R}$.

Final version. To appear in Pacific J. Math

References in corpus (1)

• The Dirichlet problem for the minimal surface equation -with possible infinite boundary data- over domains in a Riemannian surface

Cited by in corpus (1)

• A reflection principle for minimal surfaces in smooth three manifolds