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 C2 Riemannian manifolds Σ 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 C1,α 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 ⌊p2⌋. |
Notes | Preprint |
URL | https://arxiv.org/abs/2105.08135 |
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