Complete surfaces with positive extrinsic curvature in product spaces

By J. Espinar, J. A. Gálvez, H. Rosenberg. 2007

72 citations recorded by the source corpus.

We prove that every complete connected immersed surface with positive extrinsic curvature K in H 2 � 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 S 2 � R and H 2 � R are rotational spheres. Here are the key steps to achieve this. First height estimates for compactK-surfaces in a general ambient space M 2 � R with boundary in a slice are obtained. Then distance estimates for compact K-surfaces (and H -surfaces) in H 2 � R with boundary on a vertical plane are obtained. Finally we construct a quadratic form with isolated zeroes of negative index.

Subfields: Applied Mathematics, Astronomy and Astrophysics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/c1b5dd927b5c762590ddbe4cd07c7ab75b8e65a5

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In Applied Mathematics

1. A. Ros
2. P. Sicbaldi
3. W. H. Meeks
4. Joaquín Pérez
5. Barry Simon
6. D. Ruiz
7. F. Schlenk

In Astronomy and Astrophysics

1. A. Ros
2. W. H. Meeks

In Geometry and Topology

1. A. Ros
2. W. H. Meeks
3. Joaquín Pérez
4. Barry Simon

Citations in the corpus

• Geometry and Topology of some overdetermined elliptic problems (2012, background): “Rosenberg in [7] in the context of constant curvature surfaces.”