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Bi-Lipschitz Pieces between Manifolds
A well-known class of questions asks the following: If and are metric
measure spaces and is a Lipschitz mapping whose image has
positive measure, then must have large pieces on which it is bi-Lipschitz?
Building on methods of David (who is not the present author) and Semmes, we
answer this question in the affirmative for Lipschitz mappings between certain
types of Ahlfors -regular, topological -manifolds. In general, these
manifolds need not be bi-Lipschitz embeddable in any Euclidean space. To prove
the result, we use some facts on the Gromov-Hausdorff convergence of manifolds
and a topological theorem of Bonk and Kleiner. This also yields a new proof of
the uniform rectifiability of some metric manifolds.Comment: 38 page
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