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Simone Steinbrüchel

Publications and source records attributed to Simone Steinbrüchel.

4 recordsLinked to original sources
arXiv2023

Generic uniqueness for the Plateau problem

Given a complete Riemannian manifold $\mathcal{M}\subset\mathbb{R}^d$ which is a Lipschitz neighbourhood retract of dimension $m+n$, of class $C^{h,β}$ and an oriented, closed submanifold $Γ\subset \mathcal M$ of dimension $m-1$, which is a boundary in integral homology, we construct a complete metric space $\mathcal{B}$ of $C^{h,α}$-perturbations of $Γ$ inside $\mathcal{M}$, with $α<β$, 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.

math.AP↗
arXiv2021

Uniqueness of boundary tangent cones for $2$-dimensional area-minimizing currents

In this paper we show that, if $T$ is an area-minimizing $2$-dimensional integral current with $\partial T = Q [\![ Γ]\!]$, where $Γ$ is a $C^{1,α}$ curve for $α>0$ and $Q$ an arbitrary integer, then $T$ has a unique tangent cone at every boundary point, with a polynomial convergence rate. The proof is a simple reduction to the case $Q=1$, studied by Hirsch and Marini.

math.AP↗
arXiv2021

An Allard-type boundary regularity theorem for $2d$ minimizing currents at smooth curves with arbitrary multiplicity

We consider integral area-minimizing $2$-dimensional currents $T$ in $U\subset \mathbb R^{2+n}$ with $\partial T = Q[\![Γ]\!]$, where $Q\in \mathbb N \setminus \{0\}$ and $Γ$ is sufficiently smooth. We prove that, if $q\in Γ$ is a point where the density of $T$ is strictly below $\frac{Q+1}{2}$, then the current is regular at $q$. The regularity is understood in the following sense: there is a neighborhood of $q$ in which $T$ consists of a finite number of regular minimal submanifolds meeting transversally at $Γ$ (and counted with the appropriate integer multiplicity). In view of well-known examples, our result is optimal, and it is the first nontrivial generalization of a classical theorem of Allard for $Q=1$. As a corollary, if $Ω\subset \mathbb R^{2+n}$ is a bounded uniformly convex set and $Γ\subset \partial Ω$ a smooth $1$-dimensional closed submanifold, then any area-minimizing current $T$ with $\partial T = Q [\![Γ]\!]$ is regular in a neighborhood of $Γ$.

math.AP↗