The singular set of minimal surfaces near polyhedral cones

Abstract

We adapt the method of Simon [26] to prove a $C^{1,\alpha}$-regularity theorem for minimal varifolds which resemble a cone $\mathbf{C}^2_0$ over an equiangular geodesic net. For varifold classes admitting a “no-hole” condition on the singular set, we additionally establish $C^{1,\alpha}$-regularity near the cone $\mathbf{C}^2_0 \times \mathbb{R}^m$. Combined with work of Allard [2], Simon [26], Taylor [29], and Naber–Valtorta [21], our result implies a $C^{1,\alpha}$-structure for the top three strata of minimizing clusters and size-minimizing currents, and a Lipschitz structure on the $(n-3)$-stratum.