Invariant solutions for the asymptotic Plateau problem in ${\mathbb{H}}^3$

Abstract

In this paper, we present solutions to the asymptotic Plateau problem in the hyperbolic space ${\mathbb{H}}^3$. In this context, we exhibit solutions for curves that are invariant under the action of a one-parameter subgroup of isometries of ${\mathbb{H}}^3$. To achieve this, we prove the existence of foliations of ${\mathbb{H}}^3$ by minimal surfaces that are properly embedded, complete, and invariant under these subgroups, which are then used to solve the problem.