Minimal surfaces in ${\Bbb M}\sp 2\times\Bbb R$

Abstract

We study the geometry and topology of properly embedded minimal surfaces in $M\times\mathbb{R}$, where $M$ is a Riemannian surface. When $M$ is a round sphere, we give examples of all genus and we prove such minimal surfaces have exactly two ends or equal $M\times\{t\}$, for some real $t$. When $M$ has non-negative curvature, we study the conformal type of minimal surfaces in $M\times\mathbb{R}$, and we prove half-space theorems. When $M$ is the hyperbolic plane, we obtain a Jenkins-Serrin type theorem.