The singular set of minimal surfaces near polyhedral cones

Abstract

We adapt the method of Simon [JDG '93] to prove a $C^{1,\hat{I}\pm }$-regularity theorem for minimal varifolds which resemble a cone ${\mathbf{C}}_0^2$ over an equiangular geodesic net. For varifold classes admitting a "no-hole" condition on the singular set, we additionally establish $C^{1,\hat{I}\pm }$-regularity near the cone ${\mathbf{C}}_0^2\times {\mathbb{R}}^{\mathbf{m}}$. 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 $C^{1,\hat{I}\pm }$-structure for the top three strata of minimizing clusters and size-minimizing currents, and a Lipschitz structure on the $(n-3)$-stratum.

79 pages, 9 figures. Comments and suggestions are welcome!