A generalization of Reifenberg's theorem in $R^3$

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-Hölder image of the unit ball in $R^m$. In this paper we prove that a subset of $R^3$ which is well approximated by a minimal cone 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 $R^n$