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We give a sufficient condition for a metric (homology) manifold to be locally bi-Lipschitz equivalent to an open subset in R n . The condition is a Sobolev condition for a measurable coframe of flat 1-forms. In combination with an earlier work of D. Sullivan, our methods also yield an analytic characterization for smoothability of a Lipschitz manifold in terms of a Sobolev regularity for frames in a cotangent structure. In the proofs, we exploit the duality between flat chains and flat forms, and recently established differential analysis on metric measure spaces. When specialized to R n , our result gives a kind of asymptotic and Lipschitz version of the measurable Riemann mapping theorem as suggested by Sullivan.
@article{PMIHES_2011__113__1_0,
author = {Heinonen, Juha and Keith, Stephen},
title = {Flat forms, {bi-Lipschitz} parametrizations, and smoothability of manifolds},
journal = {Publications Math\'ematiques de l'IH\'ES},
pages = {1--37},
publisher = {Springer-Verlag},
volume = {113},
year = {2011},
doi = {10.1007/s10240-011-0032-4},
mrnumber = {2805596},
zbl = {1238.30039},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1007/s10240-011-0032-4/}
}
TY - JOUR AU - Heinonen, Juha AU - Keith, Stephen TI - Flat forms, bi-Lipschitz parametrizations, and smoothability of manifolds JO - Publications Mathématiques de l'IHÉS PY - 2011 SP - 1 EP - 37 VL - 113 PB - Springer-Verlag UR - http://geodesic.mathdoc.fr/articles/10.1007/s10240-011-0032-4/ DO - 10.1007/s10240-011-0032-4 LA - en ID - PMIHES_2011__113__1_0 ER -
%0 Journal Article %A Heinonen, Juha %A Keith, Stephen %T Flat forms, bi-Lipschitz parametrizations, and smoothability of manifolds %J Publications Mathématiques de l'IHÉS %D 2011 %P 1-37 %V 113 %I Springer-Verlag %U http://geodesic.mathdoc.fr/articles/10.1007/s10240-011-0032-4/ %R 10.1007/s10240-011-0032-4 %G en %F PMIHES_2011__113__1_0
Heinonen, Juha; Keith, Stephen. Flat forms, bi-Lipschitz parametrizations, and smoothability of manifolds. Publications Mathématiques de l'IHÉS, Tome 113 (2011), pp. 1-37. doi: 10.1007/s10240-011-0032-4
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