Uniqueness of area minimizing surfaces for extreme curves

Abstract

Let MM be a compact, orientable, mean convex 33-manifold with boundary ?M?M. We show that the set of all simple closed curves in ?M?M which bound unique area minimizing disks in MM is dense in the space of simple closed curves in ?M?M which are nullhomotopic in MM. We also show that the set of all simple closed curves in ?M?M which bound unique absolutely area minimizing surfaces in MM is dense in the space of simple closed curves in ?M?M which are nullhomologous in MM.