Complete surfaces with positive extrinsic curvature in product spaces

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 $S^2\times R$ 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 $M^2\times R$ 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.