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 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.

Keywords: Homogeneous product spaces, positive extrinsic curvature, Hadamard–Stoker type theorem, height estimates, classification of K-surfaces

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