Calculus of Variations and Geometric Measure Theory

C. De Lellis

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

created by delellis on 07 Nov 2021



Inserted: 7 nov 2021
Last Updated: 7 nov 2021

Year: 2021


We consider integral area-minimizing 2-dimensional currents $T$ in $U\subset \mathbb R^{2+n}$ with $\partial T = Q \llbracket \Gamma \rrbracket$, where $Q\in \mathbb N \setminus \{0\}$ and and $\Gamma$ is a sufficiently smooth curve. We prove that, if $q\in \Gamma$ 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 $\Gamma$ (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 $\Omega\subset \mathbb R^{2+n}$ is a bounded uniformly convex set and $Γ\subset \partial \Omega$ a smooth 1-dimensional closed submanifold, then any area-minimizing current T with $\partial T=Q\llbracket \Gamma \rrbracket$ is regular in a neighborhood of $\Gamma$.