Geometric and Functional Analysis

, Volume 18, Issue 4, pp 1168–1235

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

Article

DOI: 10.1007/s00039-008-0681-8

Cite this article as:
David, G., Pauw, T.D. & Toro, T. GAFA Geom. funct. anal. (2008) 18: 1168. doi:10.1007/s00039-008-0681-8

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

Keywords and phrases:

Minimal cones bi-Hölder parameterizations 

AMS Mathematics Subject Classification:

28A75 49Q05 49Q20 

Copyright information

© Birkhaeuser 2008

Authors and Affiliations

  1. 1.Université de Paris-SudOrsayFrance
  2. 2.Université Catholique de LouvainLouvain-la-NeuveBelgique
  3. 3.Department of MathematicsUniversity of WashingtonSeattleUSA

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