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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 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.
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Guy C. David, Bi-Lipschitz pieces between manifolds. Rev. Mat. Iberoam. 32 (2016), no. 1, pp. 175–218