Height estimates for surfaces with positive constant mean curvature in $\mathbb{M}^{2}\times\mathbb{R}$

Abstract

We obtain height estimates for compact embedded surfaces with positive constant mean curvature in a Riemannian product space $\mathbb{M}^{2}\times\mathbb{R}$ and boundary on a slice. We prove that these estimates are optimal for the homogeneous spaces $ℝ^3$, $\mathbb{S}^{2}\times\mathbb{R}$, and $ℍ^{2}×ℝ$ and we characterize the surfaces for which these bounds are achieved. We also give some geometric properties on properly embedded surfaces without boundary.