Generic uniqueness for the Plateau problem

TL;DR

This paper proves that for a generic perturbation of a boundary in a Riemannian manifold, there is a unique multiplicity-one solution to the Plateau problem, establishing a generic uniqueness result.

Contribution

It introduces a framework showing that for most boundary perturbations, the Plateau problem admits a unique solution in a Lipschitz neighborhood retract setting.

Findings

Generic boundary perturbations yield unique solutions

Solutions have multiplicity one

Results hold for a broad class of Riemannian manifolds

Abstract

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