Height estimates for constant mean curvature graphs in $$\mathrm {Nil}_3$$Nil3 and $${\widetilde{PSL_2}}({\mathbb {R}})$$PSL2~(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 complete graphs defined on relatively compact domains.