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

TL;DR

This paper studies the structure of area minimizing hypersurfaces modulo p in Riemannian manifolds, revealing regularity and singularity properties depending on the parity of p, with implications for understanding minimal hypersurfaces with multiplicities.

Contribution

It establishes the local structure and regularity of area minimizing hypersurfaces modulo p, extending classical results to cases with arbitrary multiplicities and moduli.

Findings

For odd p, hypersurfaces are unions of smooth minimal hypersurfaces meeting at a common boundary.

For even p, similar structure holds near points with tangent cones having an (m-1)-dimensional spine.

The paper proves uniqueness and decay towards tangent cones for all moduli.

Abstract

We consider area minimizing $m$-dimensional currents $\mathrm{mod}(p)$ 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 Leon Simon in "Cylindrical tangent cones and the singular set of minimal submanifolds" (J. Diff. Geom. 1993) in a class of 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 $\lfloor \frac{p}{2}\rfloor$.