Flat forms, bi-Lipschitz parametrizations, and smoothability of manifolds
Publications Mathématiques de l'IHÉS, Tome 113 (2011), pp. 1-37.

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.

DOI : 10.1007/s10240-011-0032-4
Heinonen, Juha 1 ; Keith, Stephen 2

1 Department of Mathematics, University of Michigan Ann Arbor, MI, 48109 USA
2 Sydney Australia
@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://archive.numdam.org/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://archive.numdam.org/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://archive.numdam.org/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. http://archive.numdam.org/articles/10.1007/s10240-011-0032-4/

[1.] Begle, E. G. Regular convergence, Duke Math. J., Volume 11 (1944), pp. 441-450 | DOI | MR | Zbl

[2.] Bishop, C. J. An A 1 weight not comparable with any quasiconformal Jacobian, In the Tradition of Ahlfors-Bers, IV (Contemp. Math., 432), Amer. Math. Soc., Providence (2007), pp. 7-18 | MR | Zbl

[3.] Bonk, M.; Kleiner, B. Quasisymmetric parametrizations of two-dimensional metric spheres, Invent. Math., Volume 150 (2002), pp. 127-183 | DOI | MR | Zbl

[4.] Bonk, M.; Lang, U. Bi-Lipschitz parameterization of surfaces, Math. Ann., Volume 327 (2003), pp. 135-169 | DOI | MR | Zbl

[5.] Bonk, M.; Heinonen, J.; Rohde, S. Doubling conformal densities, J. Reine Angew. Math., Volume 541 (2001), pp. 117-141 | DOI | MR | Zbl

[6.] Bonk, M.; Heinonen, J.; Saksman, E. The quasiconformal Jacobian problem, In the Tradition of Ahlfors and Bers, III (Contemp. Math., 355), Amer. Math. Soc., Providence (2004), pp. 77-96 | MR | Zbl

[7.] Bonk, M.; Heinonen, J.; Saksman, E. Logarithmic potentials, quasiconformal flows, and Q-curvature, Duke Math. J., Volume 142 (2008), pp. 197-239 | DOI | MR | Zbl

[8.] Borsuk, K. On some metrizations of the hyperspace of compact sets, Fundam. Math., Volume 41 (1955), pp. 168-202 | MR | Zbl

[9.] Bryant, J.; Ferry, S.; Mio, W.; Weinberger, S. Topology of homology manifolds, Ann. Math., Volume 143 (1996), pp. 435-467 | DOI | MR | Zbl

[10.] Cannon, J. W. Shrinking cell-like decompositions of manifolds. Codimension three, Ann. Math., Volume 110 (1979), pp. 83-112 | DOI | MR | Zbl

[11.] Cheeger, J. Differentiability of Lipschitz functions on metric measure spaces, Geom. Funct. Anal., Volume 9 (1999), pp. 428-517 | DOI | MR | Zbl

[12.] Connes, A.; Sullivan, D.; Teleman, N. Quasiconformal mappings, operators on Hilbert space, and local formulae for characteristic classes, Topology, Volume 33 (1994), pp. 663-681 | DOI | MR | Zbl

[13.] David, G.; Semmes, S. Fractured Fractals and Broken Dreams: Self-similar Geometry Through Metric and Measure, Oxford Lecture Series in Mathematics and its Applications, 7, Clarendon, Oxford, 1997 | MR | Zbl

[14.] Edwards, R. D. The topology of manifolds and cell-like maps, Proceedings of the International Congress of Mathematicians, Acad. Sci. Fennica, Helsinki (1980), pp. 111-127 | MR | Zbl

[15.] Evans, L. C.; Gariepy, R. F. Measure Theory and Fine Properties of Functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, 1992 | MR | Zbl

[16.] Federer, H. Geometric Measure Theory, Die Grundlehren der mathematischen Wissenschaften, 153, Springer, New York, 1969 | MR | Zbl

[17.] George Michael, A. A. On the smoothing problem, Tsukuba J. Math., Volume 25 (2001), pp. 13-45 | MR | Zbl

[18.] Gromov, M. Metric Structures for Riemannian and Non-Riemannian Spaces, Progress in Mathematics, 152, Birkhäuser, Boston, 1999 | MR | Zbl

[19.] Grove, K.; Petersen, V. P.; Wu, J. Y. Geometric finiteness theorems via controlled topology, Invent. Math., Volume 99 (1990), pp. 205-213 | DOI | MR | Zbl

[20.] Heinonen, J. Lectures on Analysis on Metric Spaces, Springer, New York, 2001 | MR | Zbl

[21.] Heinonen, J. The branch set of a quasiregular mapping, Proceedings of the International Congress of Mathematicians, vol. II, Higher Ed. Press, Beijing (2002), pp. 691-700 | MR | Zbl

[22.] Heinonen, J. Lectures on Lipschitz Analysis, Report. University of Jyväskylä Department of Mathematics and Statistics, 100, University of Jyväskylä, Jyväskylä, 2005 | MR | Zbl

[23.] Heinonen, J.; Kilpeläinen, T. BLD-mappings in W 2,2 are locally invertible, Math. Ann., Volume 318 (2000), pp. 391-396 | DOI | MR | Zbl

[24.] Heinonen, J.; Koskela, P. Quasiconformal maps in metric spaces with controlled geometry, Acta Math., Volume 181 (1998), pp. 1-61 | DOI | MR | Zbl

[25.] Heinonen, J.; Rickman, S. Geometric branched covers between generalized manifolds, Duke Math. J., Volume 113 (2002), pp. 465-529 | DOI | MR | Zbl

[26.] Heinonen, J.; Sullivan, D. On the locally branched Euclidean metric gauge, Duke Math. J., Volume 114 (2002), pp. 15-41 | DOI | MR | Zbl

[27.] Heinonen, J.; Koskela, P.; Shanmugalingam, N.; Tyson, J. T. Sobolev classes of Banach space-valued functions and quasiconformal mappings, J. Anal. Math., Volume 85 (2001), pp. 87-139 | DOI | MR | Zbl

[28.] Hirsch, M. W.; Mazur, B. Smoothings of Piecewise Linear Manifolds, Annals of Mathematics Studies, 80, Princeton University Press, Princeton, 1974 | MR | Zbl

[29.] Keith, S. Modulus and the Poincaré inequality on metric measure spaces, Math. Z., Volume 245 (2003), pp. 255-292 | DOI | MR | Zbl

[30.] Keith, S. A differentiable structure for metric measure spaces, Adv. Math., Volume 183 (2004), pp. 271-315 | DOI | MR | Zbl

[31.] Keith, S. Measurable differentiable structures and the Poincaré inequality, Indiana Univ. Math. J., Volume 53 (2004), pp. 1127-1150 | DOI | MR | Zbl

[32.] Keith, S.; Zhong, X. The Poincaré inequality is an open ended condition, Ann. Math., Volume 167 (2008), pp. 575-599 | DOI | MR | Zbl

[33.] Kirby, R. C.; Siebenmann, L. C. Foundational Essays on Topological Manifolds, Smoothings, and Triangulations, Annals of Mathematics Studies, 80, Princeton University Press, Princeton, 1977 (With notes by John Milnor and Michael Atiyah) | MR | Zbl

[34.] Koskela, P. Upper gradients and Poincaré inequalities, Lecture Notes on Analysis in Metric Spaces, Appunti Corsi Tenuti Docenti Sc. Scuola Norm. Sup., Pisa (2000), pp. 55-69 | MR | Zbl

[35.] Laakso, T. J. Plane with A -weighted metric not bi-Lipschitz embeddable to $\Bbb{R}\sp N$ , Bull. Lond. Math. Soc., Volume 34 (2002), pp. 667-676 | DOI | MR | Zbl

[36.] Luukkainen, J.; Väisälä, J. Elements of Lipschitz topology, Ann. Acad. Sci. Fenn., Ser. A 1 Math., Volume 3 (1977), pp. 85-122 | MR | Zbl

[37.] Martio, O.; Väisälä, J. Elliptic equations and maps of bounded length distortion, Math. Ann., Volume 282 (1988), pp. 423-443 | DOI | MR | Zbl

[38.] Mattila, P. Geometry of Sets and Measures in Euclidean Spaces, Cambridge Studies in Advanced Mathematics, 44, Cambridge University Press, Cambridge, 1995 | MR | Zbl

[39.] Milnor, J. Topological manifolds and smooth manifolds, Proc. Internat. Congr. Mathematicians, Inst. Mittag-Leffler, Djursholm (1963), pp. 132-138 | MR | Zbl

[40.] Milnor, J. W.; Stasheff, J. D. Characteristic Classes, Annals of Mathematics Studies, 76, Princeton University Press, Princeton, 1974 | MR | Zbl

[41.] Moise, E. E. Geometric Topology in Dimensions 2 and 3, Graduate Texts in Mathematics, 47, Springer, New York, 1977 | MR | Zbl

[42.] Müller, S.; Šverák, V. On surfaces of finite total curvature, J. Differ. Geom., Volume 42 (1995), pp. 229-258 | MR | Zbl

[43.] Munkres, J. Obstructions to imposing differentiable structures, Ill. J. Math., Volume 8 (1964), pp. 361-376 | MR | Zbl

[44.] Pugh, C. C. Smoothing a topological manifold, Topol. Appl., Volume 124 (2002), pp. 487-503 | DOI | MR | Zbl

[45.] Quinn, F. An obstruction to the resolution of homology manifolds, Mich. Math. J., Volume 34 (1987), pp. 285-291 | DOI | MR | Zbl

[46.] Reshetnyak, Y. G. Space mappings with bounded distortion, Sib. Mat. Zh., Volume 8 (1967), pp. 629-659 | MR | Zbl

[47.] Reshetnyak, Y. G. Space Mappings with Bounded Distortion, Translations of Mathematical Monographs, 73, American Mathematical Society, Providence, 1989 (Translated from the Russian by H. H. McFaden) | MR | Zbl

[48.] Rickman, S. Quasiregular Mappings, Springer, Berlin, 1993 | MR | Zbl

[49.] Semmes, S. Bi-Lipschitz mappings and strong A weights, Ann. Acad. Sci. Fenn., Ser. A 1 Math., Volume 18 (1993), pp. 211-248 | MR | Zbl

[50.] Semmes, S. Finding curves on general spaces through quantitative topology, with applications to Sobolev and Poincaré inequalities, Sel. Math., Volume 2 (1996), pp. 155-295 | DOI | MR | Zbl

[51.] Semmes, S. Good metric spaces without good parameterizations, Rev. Mat. Iberoam., Volume 12 (1996), pp. 187-275 | MR | Zbl

[52.] Semmes, S. On the nonexistence of bi-Lipschitz parameterizations and geometric problems about A -weights, Rev. Mat. Iberoam., Volume 12 (1996), pp. 337-410 | MR | Zbl

[53.] Shikata, Y. On a distance function on the set of differentiable structures, Osaka J. Math., Volume 3 (1966), pp. 65-79 | MR | Zbl

[54.] Shikata, Y. On the smoothing problem and the size of a topological manifold, Osaka J. Math., Volume 3 (1966), pp. 293-301 | MR | Zbl

[55.] Siebenmann, L.; Sullivan, D. On complexes that are Lipschitz manifolds, Geometric Topology, Academic Press, New York (1979), pp. 503-525 | MR | Zbl

[56.] Sullivan, D. Hyperbolic geometry and homeomorphisms, Geometric Topology, Academic Press, New York (1979), pp. 543-555 | MR | Zbl

[57.] Sullivan, D. Exterior d, the local degree, and smoothability, Prospects in Topology (Ann. of Math. Stud., 138), Princeton Univ. Press, Princeton (1995), pp. 328-338 | MR | Zbl

[58.] D. Sullivan, The Ahlfors-Bers measurable Riemann mapping theorem for higher dimensions. Lecture at the Ahlfors celebration, Stanford University, September 1997, http://www.msri.org/publications/ln/hosted/ahlfors/1997/sullivan/1/index.html.

[59.] Sullivan, D. On the Foundation of Geometry, Analysis, and the Differentiable Structure for Manifolds, Topics in Low-Dimensional Topology, World Sci. Publishing, River Edge (1999), pp. 89-92 | MR | Zbl

[60.] Teleman, N. The index of signature operators on Lipschitz manifolds, Inst. Hautes Études Sci. Publ. Math., Volume 58 (1984), pp. 39-78 | MR

[61.] Toro, T. Surfaces with generalized second fundamental form in L 2 are Lipschitz manifolds, J. Differ. Geom., Volume 39 (1994), pp. 65-101 | MR | Zbl

[62.] Toro, T. Geometric conditions and existence of bi-Lipschitz parameterizations, Duke Math. J., Volume 77 (1995), pp. 193-227 | DOI | MR | Zbl

[63.] Väisälä, J. Minimal mappings in Euclidean spaces, Ann. Acad. Sci. Fenn., Ser. A 1 Math., Volume 366 (1965), p. 22 | MR | Zbl

[64.] Väisälä, J. Exhaustions of John domains, Ann. Acad. Sci. Fenn., Ser. A 1 Math., Volume 19 (1994), pp. 47-57 | MR | Zbl

[65.] Whitehead, J. H. C. Manifolds with transverse fields in Euclidean space, Ann. Math., Volume 73 (1961), pp. 154-212 | DOI | MR | Zbl

[66.] Whitney, H. Geometric Integration Theory, Princeton University Press, Princeton, 1957 | MR | Zbl

[67.] K. Wildrick, Quasisymmetric structures on surfaces. Preprint, September 2007. | MR | Zbl

[68.] K. Wildrick, Quasisymmetric Parameterizations of Two-Dimensional Metric Spaces, PhD thesis, University of Michigan, 2007. | MR

Cité par Sources :