Complete surfaces with positive extrinsic curvature in product spaces

TL;DR

This paper classifies complete surfaces with positive extrinsic curvature in product spaces, showing they are properly embedded spheres or planes, and identifies the only complete constant curvature surfaces as rotational spheres.

Contribution

It provides a classification of complete surfaces with positive extrinsic curvature in $H^2\times R$ and $S^2\times R$, including height and distance estimates and the uniqueness of rotational spheres.

Findings

Complete surfaces with positive extrinsic curvature in $H^2\times R$ are properly embedded and homeomorphic to a sphere or plane.

The only complete constant extrinsic curvature surfaces in $S^2\times R$ and $H^2\times R$ are rotational spheres.

Height and distance estimates are established for compact $K$-surfaces 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 $S^2\times R$ 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.