Uniqueness of Area Minimizing Surfaces for Extreme Curves

TL;DR

This paper proves that in certain 3-manifolds, the set of boundary curves that uniquely bound minimal surfaces is dense among all nullhomotopic or nullhomologous curves, highlighting the generic nature of uniqueness.

Contribution

It establishes the density of boundary curves with unique minimal surfaces in mean convex 3-manifolds, extending understanding of minimal surface boundary behavior.

Findings

Unique area minimizing disks are dense among nullhomotopic boundary curves.

Unique absolutely area minimizing surfaces are dense among nullhomologous boundary curves.

Results apply to compact, orientable, mean convex 3-manifolds.

Abstract

Let M be a compact, orientable, mean convex 3-manifold with boundary. We show that the set of all simple closed curves in the boundary of M which bound unique area minimizing disks in M is dense in the space of simple closed curves in the boundary of M which are nullhomotopic in M. We also show that the set of all simple closed curves in the boundary of M which bound unique absolutely area minimizing surfaces in M is dense in the space of simple closed curves in the boundary of M which are nullhomologous in M.