Area minimizing hypersurfaces modulo p: A geometric free-boundary problem

We consider area minimizing m-dimensional currents mod(p) in complete C2 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 C1,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 [18] 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 ⌊p2⌋[jls-end-space/]