On doubly periodic minimal surfaces in ${\mathbb{H}}^2\times \mathbb{R}$ with finite total curvature in the quotient space

Abstract

In this paper we develop the theory of properly immersed minimal surfaces in the quotient space ${\mathbb{H}}^2\times \mathbb{R}/G,$ where $G$ is a subgroup of isometries generated by a vertical translation and a horizontal isometry in ${\mathbb{H}}^2$ without fixed points. The horizontal isometry can be either a parabolic translation along horocycles in ${\mathbb{H}}^2$ or a hyperbolic translation along a geodesic in ${\mathbb{H}}^2.$ In fact, we prove that if a properly immersed minimal surface in ${\mathbb{H}}^2\times \mathbb{R}/G$ has finite total curvature then its total curvature is a multiple of $2\pi ,$ and moreover, we understand the geometry of the ends. These theorems hold true more generally for properly immersed minimal surfaces in $M\times {\mathbb{S}}^1,$ where $M$ is a hyperbolic surface with finite topology whose ends are isometric to one of the ends of the above spaces ${\mathbb{H}}^2\times \mathbb{R}/G.$

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