Uniqueness of constant mean curvature surfaces properly immersed in a slab

Abstract

We study complete properly immersed surfaces contained in a slab of a warped product $\mathbb{R}{\times }_{\varrho }{\mathbb{P}}^2$, where ${\mathbb{P}}^2$ is complete with nonnegative Gaussian curvature. Under certain restrictions on the mean curvature of the surface we show that such an immersion does not exists or must be a leaf of the trivial totally umbilical foliation $t\in \mathbb{R}\mapsto \{t\}\times {\mathbb{P}}^2$.