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

2021/05/17 by De Lellis, Camillo, Hirsch, Jonas, Marchese, Andrea +2 · 1 citation
#49Q05 #49Q15 #49Q20 #53A10 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2105.08135

Abstract

We consider area minimizing m-dimensional currents mod(p) 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 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 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 (p)/(2)\rfloor.

Cited by

• Stationary and stable varifolds with singularities

Related

• Fine structure of the singular set of area minimizing hypersurfaces modulo p
• Co-Dimension One Area-Minimizing Currents with C^1,α Tangentially Immersed Boundary
• On the uniqueness of tangent cones to area minimizing currents at boundaries with arbitrary multiplicity
• Fine Structure of Singularities in Area-Minimizing Currents Mod(q)
• Homologically area-minimizing surfaces mod v have at worst codimension 2 singular sets asymptotically