2013/05/21 by Laurent Hauswirth, Hauswirth, Laurent, Ana Menezes +1
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Point processes and geometric inequalities
paper · pdf · doi:10.48550/arxiv.1305.4813
openalex publication_date 2013/05/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we develop the theory of properly immersed minimal surfaces in the quotient space \mathbb H2×\mathbb R/G, where G is a subgroup of isometries generated by a vertical translation and a horizontal isometry in \mathbb H2 without fixed points. The horizontal isometry can be either a parabolic translation along horocycles in \mathbb H2 or a hyperbolic translation along a geodesic in \mathbb H2. In fact, we prove that if a properly immersed minimal surface in \mathbb H2×\mathbb R/G has finite total curvature then its total curvature is a multiple of 2π, and moreover, we understand the geometry of the ends. These theorems hold true more generally for properly immersed minimal surfaces in M×\mathbb S1, where M is a hyperbolic surface with finite topology whose ends are isometric to one of the ends of the above spaces \mathbb H2×\mathbb R/G.