Generic uniqueness of least area planes in hyperbolic space

Abstract

We study the number of solutions of the asymptotic Plateau problem in ${ℍ}^3$. By using the analytical results in our previous paper, and some topological arguments, we show that there exists an open dense subset of $C^3$ Jordan curves in $S_{\infty }^2\left({ℍ}^3\right)$ such that any curve in this set bounds a unique least area plane in ${ℍ}^3$.