The singular set of minimal surfaces near polyhedral cones

We adapt the method of Simon [JDG '93] to prove a $C^{1,\alpha }$-regularity theorem for minimal varifolds which resemble a cone ${\mathbf{C}}_{\mathbf{0}}^{\mathbf{2}}$ 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 ${\mathbf{C}}_{\mathbf{0}}^{\mathbf{2}}\times {\mathbb{R}}^{\mathbf{m}}$. Combined with work of Allard [Ann. of Math. '72], Simon [JDG '93], Taylor [Ann. of Math. '76], and Naber-Valtorta [Ann. of Math. '17], 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.