2007/05/04 by Jose M. Espinar, José M. Espinar, Espinar, Jose M. +5
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry and complex manifolds #math.DG
paper · pdf · doi:10.48550/arxiv.0705.0585
28 pages
openalex publication_date 2007/05/04 · arxiv created 2007/05/09 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove that every complete connected immersed surface with positive extrinsic curvature K in H2× R must be properly embedded, homeomorphic to a sphere or a plane and, in the latter case, study the behavior of the end. Then, we focus our attention on surfaces with positive constant extrinsic curvature (K-surfaces). We establish that the only complete K-surfaces in S2× R and H2× R are rotational spheres. Here are the key steps to achieve this. First height estimates for compact K-surfaces in a general ambient space M2× R with boundary in a slice are obtained. Then distance estimates for compact K-surfaces (and H-surfaces) in H2× R with boundary on a vertical plane are obtained. Finally we construct a quadratic form with isolated zeroes of negative index.