Complete Constant Mean Curvature surfaces and Bernstein type Theorems in $\mathbb{M}^2\times \mathbb{R}$
| dc.creator | Espinar, Jose M. | |
| dc.creator | Rosenberg, Harold | |
| dc.date | 2008-08-25 | |
| dc.date.accessioned | 2026-07-07T09:58:25Z | |
| dc.date.available | 2026-07-07T09:58:25Z | |
| dc.description | In this paper we study constant mean curvature surfaces $Σ$ 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 $Σ$. We classify these surfaces according to the infimum $c(Σ)$ of the Gaussian curvature of the projection of $Σ$. When $H \neq 0$ and $c(Σ)\geq 0$, then $Σ$ is a cylinder over a complete curve with curvature 2H. If H=0 and $c(Σ) \geq 0$, then $Σ$ must be a vertical plane or $Σ$ is a slice $\mathbb{M}^2 \times {t}$, or $\mathbb{M}^2 \equiv \mathbb{R}^2$ with the flat metric and $Σ$ is a tilted plane (after possibly passing to a covering space). When $c(Σ)<0$ and $H>\sqrt{-c(Σ)} /2$, then $Σ$ 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 $c(\mathbb{M}^2)<0$. | |
| dc.identifier | https://arxiv.org/abs/0808.3412 | |
| dc.identifier | http://arxiv.org/abs/0808.3412 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/167706 | |
| dc.subject | Differential Geometry | |
| dc.title | Complete Constant Mean Curvature surfaces and Bernstein type Theorems in $\mathbb{M}^2\times \mathbb{R}$ | |
| dc.type | text |