# Complete surfaces with positive extrinsic curvature in product spaces

### José M. Espinar

Universidad de Granada, Spain### José A. Gálvez

Universidad de Granada, Spain### Harold Rosenberg

Rio de Janeiro, Brazil

## Abstract

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 \mathbb{S}2 × 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 \mathbb{M}2 × 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.

## Cite this article

José M. Espinar, José A. Gálvez, Harold Rosenberg, Complete surfaces with positive extrinsic curvature in product spaces. Comment. Math. Helv. 84 (2009), no. 2, pp. 351–386

DOI 10.4171/CMH/165