Calculus of Variations and Geometric Measure Theory

M. Colombo - N. Edelen - L. Spolaor

The singular set of minimal surfaces near polyhedral cones

created by spolaor on 05 Oct 2017
modified on 13 Dec 2019


Accepted Paper

Inserted: 5 oct 2017
Last Updated: 13 dec 2019

Journal: JDG
Year: 2017


We adapt the method of Simon to prove a $C^{1,\alpha}$-regularity theorem for minimal varifolds which resemble a cone $\bf C_0^2$ over an equiangular geodesic net. For varifold classes admitting a ``no-hole'' condition on the singular set, we additionally establish $C^{1,\alpha}$-regularity near the cone $\bf C_0^2 \times \mathbb R^m$. Combined with work of Allard, Simon, Taylor, and Naber-Valtorta, 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.