The singular set of minimal surfaces near polyhedral cones

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