TitleArea minimizing hypersurfaces modulo p: a geometric free-boundary problem
Publication TypeJournal Article
Year of Publication2021
AuthorsDe Lellis C, Hirsch J, Marchese A, Spolaor L, Stuvard S
Type of ArticleInterior 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 m1, and the result is optimal. For even p such structure holds in a neighborhood of any point where at least one tangent cone has (m1)-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

URLhttps://arxiv.org/abs/2105.08135
Order: 
11