Height and Area Estimates for Constant Mean Curvature Graphs in E(κ,τ)-Spaces

We obtain area growth estimates for constant mean curvature graphs in E ( κ , τ ) -spaces with κ ≤ 0 , by finding sharp upper bounds for the volume of geodesic balls in E ( κ , τ ) . We focus on complete graphs and graphs with zero boundary values. For instance, we prove that entire graphs in E ( κ...

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Bibliographic Details
Published in:The Journal of geometric analysis 2017-10, Vol.27 (4), p.3441-3473
Main Authors: Manzano, José M., Nelli, Barbara
Format: Article
Language:English
Subjects:
Abstract Harmonic Analysis
Convex and Discrete Geometry
Curvature
Differential Geometry
Dynamical Systems and Ergodic Theory
Fourier Analysis
Geometry
Global Analysis and Analysis on Manifolds
Graphs
Mathematics
Mathematics and Statistics
Upper bounds
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Summary:We obtain area growth estimates for constant mean curvature graphs in E ( κ , τ ) -spaces with κ ≤ 0 , by finding sharp upper bounds for the volume of geodesic balls in E ( κ , τ ) . We focus on complete graphs and graphs with zero boundary values. For instance, we prove that entire graphs in E ( κ , τ ) 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 the Heisenberg space ( κ = 0 ), and prove a Collin–Krust type estimate for such minimal graphs.
ISSN:1050-6926
1559-002X
DOI:10.1007/s12220-017-9810-7