Height and area estimates for constant mean curvature graphs in E(κ,τ)-spaces

Abstract

We obtain area growth estimates for constant mean curvature graphs in $\mathbb{E}(\kappa ,\tau )$-spaces with $\kappa \le 0$, 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.

24 pages, 1 table. Some typos have been corrected and Corollary 4 has been added

References in corpus (2)

• On the classification of Killing submersions and their isometries
• On complete constant mean curvature vertical multigraphs in E(κ,τ)

Cited by in corpus (7)

• Dual quadratic differentials and entire minimal graphs in Heisenberg space
• A duality for prescribed mean curvature graphs in Riemannian and Lorentzian Killing submersions
• Invariant constant mean curvature tubes around a horizontal geodesic in $\mathbb{E}(\kappa ,\tau )$-spaces
• Conjugate Plateau constructions in product spaces
• Entire Constant Mean Curvature Graphs in ${\mathbb{H}}^2\times \mathbb{R}$
• Parabolicity of invariant Surfaces
• Minimal graphs over non-compact domains in 3-manifolds fibered by a Killing vector field