TitleAn Allard-type boundary regularity theorem for 2d minimizing currents at smooth curves with arbitrary multiplicity
Publication TypeJournal Article
Year of Publication2021
AuthorsDe Lellis C, Nardulli S, Steinbruechel S
Type of ArticleBoundary
Abstract

We consider integral area-minimizing 2-dimensional currents T in UR2+n with T=Q\aΓ, where QN{0} and Γ is sufficiently smooth. We prove that, if qΓ is a point where the density of T is strictly below Q+12, 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 ΩR2+n is a bounded uniformly convex set and ΓΩ a smooth 1-dimensional closed submanifold, then any area-minimizing current T with T=Q\aΓ is regular in a neighborhood of Γ.

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