Asymptotic $H$–Plateau problem in $\mathbb{H}^3$

Abstract

We show that for any Jordan curve $\Gamma$ in $S_{\infty }^2\left({ℍ}^3\right)$ with at least one smooth point, there exists an embedded $H$–plane ${\mathcal{P}}_H$ in ${ℍ}^3$ with ${\partial }_{\infty }{\mathcal{P}}_H=\Gamma$ for any $H\in \left[0,1\right)$.