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A generalization of Reifenberg’s theorem in $${\mathbb{R}}^3$$. (English) Zbl 1169.49040
E. R. Reifenberg [Acta Math. 104, 1–92 (1960; Zbl 0099.08503)] formulated the Plateau problem for $$m$$-dimensional surfaces of varying topological type in $$\mathbb R^ k$$ and proved that a given set $$\Gamma\subset\mathbb R^ k$$ homeomorphic to $$\mathbb S^{m-1}$$ bounds a set $$\Sigma_ 0$$ which minimizes the $$\mathcal H^ m$$ Hausdorff measure among all sets in the appropriate class and he showed that for almost every $$x\in\Sigma_ 0$$ there exists a neighborhood of $$x$$ which is a topological disk of dimension $$m$$. Finally, it was shown that if a set is close to an $$m$$-plane in the Hausdorff distance sense at all points and at all scales, then it is locally bi-Hölder equivalent to a ball of $$\mathbb R^ m$$.
In this paper, the authors 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 scale is locally a bi-Hölder deformation of a minimal cone. Also, they prove an analogous result for more general cones in $$\mathbb R^ n$$.

##### MSC:
 49Q05 Minimal surfaces and optimization 49Q20 Variational problems in a geometric measure-theoretic setting 28A75 Length, area, volume, other geometric measure theory
##### Keywords:
minimal cones; bi-Hölder parameterizations
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