A Generalization Of Reifenberg’s Theorem In $${\mathbb{R}}^3$$

Abstract

In 1960 Reifenberg proved the topological disc property. He showed that a subset of R-n which is well approximated by m-dimensional affine spaces at each point and at each (small) scale is locally a bi-Holder image of the unit ball in R-m. In this paper we prove that a subset of R-3 which is well approximated in the Hausdorff distance sense by one of the three standard area-minimizing cones at each point and at each (small) scale is locally a bi-Holder deformation of a minimal cone. We also prove an analogous result for more general cones in R-n.