Title | An Allard-type boundary regularity theorem for 2d minimizing currents at smooth curves with arbitrary multiplicity |
Publication Type | Journal Article |
Year of Publication | 2021 |
Authors | De Lellis C, Nardulli S, Steinbruechel S |
Type of Article | Boundary |
Abstract | We consider integral area-minimizing 2-dimensional currents T in U⊂R2+n with ∂T=Q\aΓ, where Q∈N∖{0} and Γ is sufficiently smooth. We prove that, if q∈Γ 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 Ω⊂R2+n is a bounded uniformly convex set and Γ⊂∂Ω a smooth 1-dimensional closed submanifold, then any area-minimizing current T with ∂T=Q\aΓ is regular in a neighborhood of Γ. |
Notes | Preprint |
Order:
4