Generic Uniqueness of Area Minimizing Disks for Extreme Curves

TL;DR

This paper proves that for most simple closed curves on the boundary of certain 3-manifolds, there exists a unique embedded area-minimizing disk or surface spanning the curve, ensuring uniqueness in generic cases.

Contribution

It establishes the generic uniqueness of area-minimizing disks and surfaces for nullhomotopic curves in mean convex 3-manifolds with trivial second homology.

Findings

Unique area-minimizing disks for generic nullhomotopic curves

Unique absolutely area-minimizing surfaces for generic nullhomotopic curves

Results apply to boundary curves in mean convex 3-manifolds with trivial second homology

Abstract

We show that for a generic nullhomotopic simple closed curve C in the boundary of a compact, orientable, mean convex 3-manifold M with trivial second homology, there is a unique area minimizing disk D embedded in M where the boundary of D is C. We also show that the same is true for absolutely area minimizing surfaces.