Reifenberg Parameterizations for Sets with Holes

Details Reifenberg Parameterizations for Sets with Holes Reifenberg Parameterizations for Sets with Holes

2009-10-26

arxiv.org/abs/0910.4869v1

Mathematics

Classical Analysis and ODEs

95 pages, 3 figures

Scientific paper

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 $\R^d$.

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