Complete surfaces with positive extrinsic curvature in product spaces

Abstract

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