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 ${\mathbb{R}}^n$ which is well approximated by m-dimensional affine spaces at each point and at each (small) scale is locally a bi-Hölder image of the unit ball in ${\mathbb{R}}^m$. In this paper we prove that a subset of ${\mathbb{R}}^3$ 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-Hölder deformation of a minimal cone. We also prove an analogous result for more general cones in ${\mathbb{R}}^n$.