#### Vol. 316, No. 2, 2022

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

### Abigail Folha and Harold Rosenberg

Vol. 316 (2022), No. 2, 307–333
##### Abstract

For $0\le H<\frac{1}{2}$, we construct entire $H$-graphs in ${ℍ}^{2}×ℝ$ that are parabolic and not invariant by one parameter groups of isometries of ${ℍ}^{2}×ℝ$. Their asymptotic boundaries are $\left({\partial }_{\infty }{ℍ}^{2}\right)×ℝ$; they are dense at infinity. Previously, the only known examples of entire $H$-graphs, $0, 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
Revised: 14 July 2020
Accepted: 1 November 2021
Published: 6 April 2022
##### Authors
 Abigail Folha Instituto de Matemática e Estatística Universidade Federal Fluminense Niterói Brazil Harold Rosenberg Instituto Nacional de Matemática Pura e Aplicada Rio de Janeiro Brazil