### Abstract

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

Original language | English (US) |
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Pages (from-to) | 1491-1512 |

Number of pages | 22 |

Journal | Annali di Matematica Pura ed Applicata |

Volume | 195 |

Issue number | 5 |

DOIs | |

State | Published - Oct 1 2016 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Applied Mathematics

### Keywords

- Finite total curvature
- Holomorphic quadratic differential
- Minimal surfaces