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

TL;DR

This paper establishes sharp area growth estimates for constant mean curvature graphs in $ ext{E}( ext{kappa}, au)$-spaces with non-positive curvature, providing bounds for both intrinsic and extrinsic areas, and analyzing special examples and minimal graphs.

Contribution

It introduces new sharp bounds for area growth of constant mean curvature graphs in $ ext{E}( ext{kappa}, au)$-spaces, including entire and boundary-zero graphs, and explores specific examples like invariant surfaces and minimal graphs.

Findings

Entire graphs with critical mean curvature have at most cubic intrinsic area growth.

Sharp upper bounds are obtained for extrinsic area growth of graphs with zero boundary values.

A relation between height and area growth for minimal graphs in Heisenberg space is established.

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.