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

We consider area minimizing m -dimensional currents mod ( p ) in complete C 2 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 C 1 , 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 [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 ⌊ p 2 ⌋ .