Generic regularity for minimizing hypersurfaces in dimension 11

TL;DR

This paper proves that area-minimizing hypersurfaces are generically smooth in 11 dimensions and have controlled singular sets in higher dimensions after small perturbations, advancing understanding of regularity in geometric measure theory.

Contribution

It establishes generic smoothness of minimizing hypersurfaces in dimension 11 and bounds on singular sets in higher dimensions under small perturbations.

Findings

Hypersurfaces are generically smooth in dimension 11.

Singular sets in higher dimensions are limited in size after perturbations.

Provides new regularity results in geometric measure theory.

Abstract

We prove that area-minimizing hypersurfaces are generically smooth in ambient dimension $11$ in the context of the Plateau problem and of area minimization in integral homology. For higher ambient dimensions, $n+1 \geq 12$, we prove in the same two contexts that area-minimizing hypersurfaces have at most an $n-10-\epsilon_n$ dimensional singular set after an arbitrarily $C^\infty$-small perturbation of the Plateau boundary or the ambient Riemannian metric, respectively.