Entire constant mean curvature graphs in $\mathbb{H}^2\times\mathbb{R}$

Abigail Folha and Harold Rosenberg

Vol. 316 (2022), No. 2, 307–333

DOI: 10.2140/pjm.2022.316.307

Abstract

For $0\le H<\frac{1}{2}$, we construct entire $H$-graphs in ${ℍ}^2\times ℝ$ that are parabolic and not invariant by one parameter groups of isometries of ${ℍ}^2\times ℝ$. Their asymptotic boundaries are $({\partial }_{\infty }{ℍ}^2)\times ℝ$; they are dense at infinity. Previously, the only known examples of entire $H$-graphs, $0<H<\frac{1}{2}$, were conformally hyperbolic invariant surfaces. When $H=0$, the examples are minimal graphs constructed by P. Collin and the second author.

Keywords

entire graphs, constant mean curvature, parabolic

Mathematical Subject Classification 2010

Primary: 53A10

Secondary: 53C21, 53C42

Milestones

Received: 17 June 2019

Revised: 14 July 2020

Accepted: 1 November 2021

Published: 6 April 2022