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

Ana Menezes

Vol. 268 (2014), No. 1, 155–172

DOI: 10.2140/pjm.2014.268.155

Abstract

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

Keywords

constant mean curvature surface, periodic surface, Alexandrov reflection

Mathematical Subject Classification 2010

Primary: 53A10, 53C42

Milestones

Received: 4 June 2012

Revised: 9 September 2013

Accepted: 16 September 2013

Published: 21 May 2014