In this paper, we develop the theory of properly immersed minimal surfaces in the quotient space (H2× R) / G, where G is a subgroup of isometries generated by a vertical translation and a horizontal isometry (without fixed points) in H2. The horizontal isometry can be either a parabolic translation along horocycles in H2 or a hyperbolic translation along a geodesic in H2. We prove that if a properly immersed minimal surface in (H2× 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× S1, where M is a hyperbolic surface with finite topology whose ends are isometric to one of the ends of the above spaces (H2× R) / G.
All Science Journal Classification (ASJC) codes
- Applied Mathematics
- Finite total curvature
- Holomorphic quadratic differential
- Minimal surfaces