Entire Constant Mean Curvature Graphs in $\mathbb{H}^2\times\mathbb{R}$

TL;DR

This paper constructs new entire constant mean curvature graphs in hyperbolic space times the real line, exhibiting dense asymptotic boundaries and non-invariance under isometries, extending previous minimal surface results.

Contribution

It introduces a method to construct entire H-graphs in  with specific asymptotic and symmetry properties for 0  H < 1/2.

Findings

Existence of entire H-graphs with dense asymptotic boundaries

Construction of non-invariant parabolic graphs in 

Extension of minimal graph results to constant mean curvature cases

Abstract

For $0\leq H< 1/2$, we construct entire $H$-graphs in $\mathbb{H}^2\times\mathbb{R}$ that are parabolic and not invariant by one parameter groups of isometries of $\mathbb{H}^2\times\mathbb{R}$. Their asymptotic boundaries are $(\partial_\infty\mathbb{H}^2)\times\mathbb{R}$; they are dense at infinity. When $H=0$ the examples are minimal graphs constructed by P. Collin and the second author [2].