Complete surfaces with positive extrinsic curvature in product spaces

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.