Reifenberg Parameterizations for Sets with Holes

By G. David, T. Toro. 2009

102 citations recorded by the source corpus.

We extend the proof of Reifenberg's Topological Disk Theorem to allow the case of sets with holes, and give sufficient conditions on a set $E$ for the existence of a bi-Lipschitz parameterization of $E$ by a $d$-dimensional plane or smooth manifold. Such a condition is expressed in terms of square summability for the P. Jones numbers $\beta_1(x,r)$. In particular, it applies in the locally Ahlfors-regular case to provide very big pieces of bi-Lipschitz images of $\R^d$.

View Paper: https://www.semanticscholar.org/paper/90e1877e39562e7c56b29ade1521bdd9baf4227c

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Citations in the corpus

Rectifiable-Reifenberg and the Regularity of Stationary and Minimizing Harmonic Maps (2015, methodology): “s worth mentioning that some interesting generalizations of Reifenberg’s theorem have been proved in numorous sources. Generalizations in the spirit of this paper are explored in [DS93], [Tor95], and [DT12]. However, in each of these cases the requirements of these theorems are too stringent to be a”
The singular structure and regularity of stationary varifolds (2015, methodology): “set S is contained near a subspace, not conversely that the subspace is also contained near S. In exchange, we will only conclude the set is rectiﬁable. A result in this direction was ﬁr st proved in [DT12], but again the hypotheses are too restrictive for the applications of this paper, and additio”
An analyst’s traveling salesman theorem for sets of dimension larger than one (2016, methodology): “This is the main result as stated, but there are special properties that are implicit in David and Toro’s version that we shall employ.”