, Volume 18, Issue 4, pp 1168-1235
Date: 06 Nov 2008

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

Rent the article at a discount

Rent now

* Final gross prices may vary according to local VAT.

Get Access

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 \({\mathbb{R}}^n\) .

T.T. partially supported by the NSF under Grant DMS-0244834.
Received: July 2006, Revised: August 2007, Accepted: January 2008