Number of Least area planes in gromov hyperbolic 3-spaces

Abstract

We show that for a generic simple closed curve Γ in the asymptotic boundary of a Gromov hyperbolic 3-space with cocompact metric X, there exists a unique least area plane Σ in X such that ∂∞Σ = Γ. This result has interesting topological applications for constructions of canonical 2-dimensional objects in Gromov hyperbolic 3-manifolds.