# Commentarii Mathematici Helvetici

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**Volume 84, Issue 2, 2009, pp. 351–386**

**DOI: 10.4171/CMH/165**

Published online: 2009-06-30

Complete surfaces with positive extrinsic curvature in product spaces

José M. Espinar^{[1]}, José A. Gálvez

^{[2]}and Harold Rosenberg

^{[3]}(1) Universidad de Granada, Spain

(2) Universidad de Granada, Spain

(3) Rio de Janeiro, Brazil

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
\mathbb{S}^{2} × ℝ and ℍ^{2} × 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 ℍ^{2} × ℝ with boundary on a
vertical plane are obtained.
Finally we construct a quadratic form
with isolated zeroes of negative index.

*Keywords: *Homogeneous product spaces, positive extrinsic curvature, Hadamard–Stoker type theorem, height estimates, classification of `K -surfaces
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Espinar José, Gálvez José, Rosenberg Harold: Complete surfaces with positive extrinsic curvature in product spaces. *Comment. Math. Helv.* 84 (2009), 351-386. doi: 10.4171/CMH/165