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

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 $\nu = \meta{N}{\partial_t}$ 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 $H \neq 0$ and $c(\Sigma)\geq 0$, then $\Sigma$ is a cylinder over a complete curve with curvature 2H. If H=0 and $c(\Sigma) \geq 0$, 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 $c(\mathbb{M}^2)<0$.