Height and area estimates for constant mean curvature graphs in E(\kappa,\tau)-spaces

Abstract

We obtain area growth estimates for constant mean curvature graphs in $\mathbb{E}(\kappa,\tau)$-spaces with $\kappa\leq 0$, by finding sharp upper bounds for the volume of geodesic balls in $\mathbb{E}(\kappa,\tau)$. We focus on complete graphs and graphs with zero boundary values. For instance, we prove that entire graphs in $\mathbb{E}(\kappa,\tau)$ with critical mean curvature have at most cubic intrinsic area growth. We also obtain sharp upper bounds for the extrinsic area growth of graphs with zero boundary values, and study distinguished examples in detail such as invariant surfaces, $k$-noids and ideal Scherk graphs. Finally we give a relation between height and area growth of minimal graphs in Heisenberg space ($\kappa=0$), and prove a Collin-Krust type estimate for such minimal graphs.