Number of Least Area Planes in Gromov Hyperbolic 3-Spaces

TL;DR

This paper proves that for most simple closed curves at infinity in certain hyperbolic 3-spaces, there is a unique least area plane spanning each curve, with implications for 3-manifold topology.

Contribution

It establishes the generic uniqueness of least area planes in Gromov hyperbolic 3-spaces with cocompact metric, advancing understanding of minimal surfaces in hyperbolic geometry.

Findings

Unique least area plane for generic boundary curves

Topological applications in 3-manifold constructions

Extension of minimal surface theory in hyperbolic spaces

Abstract

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