## Abstract

Let Σ be a smooth Riemannian manifold, Γ ⊂Σ a smooth closed oriented submanifold of codimension higher than 2 and T an integral area-minimizing current in Σ which bounds Γ. We prove that the set of regular points of T at the boundary is dense in Γ. Prior to our theorem the existence of any regular point was not known, except for some special choice of Σ and Γ. As a corollary of our theorem we answer to a question in Almgren's Almgren's big regularity paper from 2000 showing that, if Γ is connected, then T has at least one point p of multiplicity 1 2 , namely there is a neighborhood of the point p where T is a classical submanifold with boundary Γ; we generalize Almgren's connectivity theorem showing that the support of T is always connected if Γ is connected; we conclude a structural result on T when Γ consists of more than one connected component, generalizing a previous theorem proved by Hardt and Simon in 1979 when Σ = Rm+1 and T is m-dimensional.

Original language | English (US) |
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Journal | Memoirs of the American Mathematical Society |

Volume | 291 |

Issue number | 1446 |

DOIs | |

State | Published - 2023 |

## Keywords

- area-minimizing currents
- Boundary regularity
- calculus of variations
- geometric measure theory
- minimal surfaces

## ASJC Scopus subject areas

- General Mathematics
- Applied Mathematics