Title | Area minimizing hypersurfaces modulo $p$: a geometric free-boundary problem |

Publication Type | Journal Article |

Year of Publication | 2021 |

Authors | De Lellis C, Hirsch J, Marchese A, Spolaor L, Stuvard S |

Type of Article | Interior regularity |

Abstract | We consider area minimizing $m$-dimensional currents $\modp$ in complete $C^2$ Riemannian manifolds $\Sigma$ of dimension $m+1$. For odd moduli we prove that, away from a closed rectifiable set of codimension $2$, the current in question is, locally, the union of finitely many smooth minimal hypersurfaces coming together at a common $C^{1,\alpha}$ boundary of dimension $m-1$, and the result is optimal. For even $p$ such structure holds in a neighborhood of any point where at least one tangent cone has $(m-1)$-dimensional spine. These structural results are indeed the byproduct of a theorem that proves (for any modulus) uniqueness and decay towards such tangent cones. The underlying strategy of the proof is inspired by the techniques developed by Simon in \cite{Simon} in a class of \emph{multiplicity one} stationary varifolds. The major difficulty in our setting is produced by the fact that the cones and surfaces under investigation have arbitrary multiplicities ranging from $1$ to $\lfloor \frac{p}{2}\rfloor$. |

Notes | Preprint |

URL | https://arxiv.org/abs/2105.08135 |

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