Generic uniqueness for the Plateau problem

Abstract

Given a complete Riemannian manifold $M\subset R^d$ which is a Lipschitz neighborhood retract of dimension $m+n$, of class $C^{h,\beta }$ and an oriented, closed submanifold $\Gamma \subset M$ of dimension $m-1$, which is a boundary in integral homology, we construct a complete metric space $B$ of $C^{h,\alpha }$-perturbations of Γ inside $M$, with $\alpha <\beta$, enjoying the following property. For the typical element $b\in B$, in the sense of Baire categories, there exists a unique m-dimensional integral current in $M$ which solves the corresponding Plateau problem and it has multiplicity one.