Complete Constant Mean Curvature surfaces and Bernstein type Theorems in ${\mathbb{M}}^2\times \mathbb{R}$

Abstract

In this paper we study constant mean curvature surfaces $\Sigma$ in a product space, ${\mathbb{M}}^2\times \mathbb{R}$, where ${\mathbb{M}}^2$ is a complete Riemannian manifold. We assume the angle function $ν= \meta{N}{\partial_t}$ does not change sign on $\Sigma$. We classify these surfaces according to the infimum $c(\Sigma )$ of the Gaussian curvature of the projection of $\Sigma$. When $H\ne 0$ and $c(\Sigma )\ge 0$, then $\Sigma$ is a cylinder over a complete curve with curvature 2H. If H=0 and $c(\Sigma )\ge 0$, then $\Sigma$ must be a vertical plane or $\Sigma$ is a slice ${\mathbb{M}}^2\times t$, or ${\mathbb{M}}^2\equiv {\mathbb{R}}^2$ with the flat metric and $\Sigma$ is a tilted plane (after possibly passing to a covering space). When $c(\Sigma )<0$ and $H>\sqrt{-c(\Sigma )}/2$, then $\Sigma$ is a vertical cylinder over a complete curve of ${\mathbb{M}}^2$ of constant geodesic curvature $2H$. This result is optimal. We also prove a non-existence result concerning complete multi-graphs in ${\mathbb{M}}^2\times \mathbb{R}$, when $c({\mathbb{M}}^2)<0$.