Uniqueness of area minimizing surfaces for extreme curves

Abstract

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