Calculus of Variations and Geometric Measure Theory

G. Caldini - A. Marchese - A. Merlo - S. Steinbr├╝chel

Generic uniqueness for the Plateau problem

created by marchese on 01 Feb 2023
modified on 28 Aug 2023


Accepted Paper

Inserted: 1 feb 2023
Last Updated: 28 aug 2023

Journal: J. Math. Pures Appl.
Year: 2023


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.