Bi-Lipschitz pieces between manifolds

• Autores: Guy David
• Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 32, Nº 1, 2016, págs. 175-218
• Idioma: inglés
• DOI: 10.4171/RMI/883
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• Resumen

  • A well-known class of questions asks the following: if X and Y are metric measure spaces and f:X→Y is a Lipschitz mapping whose image has positive measure, then must f 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 s-regular, topological d-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.