Height estimates for constant mean curvature graphs in ${\mathrm{Nil}}_3$ and ${\widetilde{PSL}}_2(\mathbb{R})$

Abstract

In this paper we obtain height estimates for compact, constant mean curvature vertical graphs in the homogeneous spaces ${\mathrm{Nil}}_3$ and ${\widetilde{PSL}}_2(\mathbb{R})$. As a straightforward consequence, we announce a structure-type result for proper graphs defined on relatively compact domains.