Geometria Complessa e Geometria Differenziale
Geometria Complessa e Geometria Differenziale
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G. Caldini - A. Marchese - A. Merlo - S. Steinbruechel

Generic uniqueness for the Plateau problem

created by caldini on 03 Feb 2023
modified on 31 Aug 2023


Accepted Paper

Inserted: 3 feb 2023
Last Updated: 31 aug 2023

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

ArXiv: 2302.01320 PDF

Comments are welcome!


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.


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