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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
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