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Flat forms, bi-Lipschitz parametrizations, and smoothability of manifolds

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Abstract

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.

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References

  1. E. G. Begle, Regular convergence, Duke Math. J., 11 (1944), 441–450.

    Article  MathSciNet  MATH  Google Scholar 

  2. C. J. Bishop, An A 1 weight not comparable with any quasiconformal Jacobian, in In the Tradition of Ahlfors-Bers, IV, Contemp. Math., vol. 432, pp. 7–18, Amer. Math. Soc., Providence, 2007.

    Google Scholar 

  3. M. Bonk and B. Kleiner, Quasisymmetric parametrizations of two-dimensional metric spheres, Invent. Math., 150 (2002), 127–183.

    Article  MathSciNet  MATH  Google Scholar 

  4. M. Bonk and U. Lang, Bi-Lipschitz parameterization of surfaces, Math. Ann., 327 (2003), 135–169.

    Article  MathSciNet  MATH  Google Scholar 

  5. M. Bonk, J. Heinonen, and S. Rohde, Doubling conformal densities, J. Reine Angew. Math., 541 (2001), 117–141.

    Article  MathSciNet  MATH  Google Scholar 

  6. M. Bonk, J. Heinonen, and E. Saksman, The quasiconformal Jacobian problem, in In the Tradition of Ahlfors and Bers, III, Contemp. Math., vol. 355, pp. 77–96, Amer. Math. Soc., Providence, 2004.

    Google Scholar 

  7. M. Bonk, J. Heinonen, and E. Saksman, Logarithmic potentials, quasiconformal flows, and Q-curvature, Duke Math. J., 142 (2008), 197–239.

    Article  MathSciNet  MATH  Google Scholar 

  8. K. Borsuk, On some metrizations of the hyperspace of compact sets, Fundam. Math., 41 (1955), 168–202.

    MathSciNet  MATH  Google Scholar 

  9. J. Bryant, S. Ferry, W. Mio, and S. Weinberger, Topology of homology manifolds, Ann. Math., 143 (1996), 435–467.

    Article  MathSciNet  MATH  Google Scholar 

  10. J. W. Cannon, Shrinking cell-like decompositions of manifolds. Codimension three, Ann. Math., 110 (1979), 83–112.

    Article  MathSciNet  Google Scholar 

  11. J. Cheeger, Differentiability of Lipschitz functions on metric measure spaces, Geom. Funct. Anal., 9 (1999), 428–517.

    Article  MathSciNet  MATH  Google Scholar 

  12. A. Connes, D. Sullivan, and N. Teleman, Quasiconformal mappings, operators on Hilbert space, and local formulae for characteristic classes, Topology, 33 (1994), 663–681.

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  14. R. D. Edwards, The topology of manifolds and cell-like maps, in Proceedings of the International Congress of Mathematicians (Helsinki, 1978), pp. 111–127, Acad. Sci. Fennica, Helsinki, 1980.

    Google Scholar 

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

    MATH  Google Scholar 

  16. H. Federer, Geometric Measure Theory, Die Grundlehren der mathematischen Wissenschaften, vol. 153, Springer, New York, 1969.

    MATH  Google Scholar 

  17. A. A. George Michael, On the smoothing problem, Tsukuba J. Math., 25 (2001), 13–45.

    MathSciNet  MATH  Google Scholar 

  18. M. Gromov, Metric Structures for Riemannian and Non-Riemannian Spaces, Progress in Mathematics, vol. 152, Birkhäuser, Boston, 1999.

    MATH  Google Scholar 

  19. K. Grove, V. P. Petersen, and J. Y. Wu, Geometric finiteness theorems via controlled topology, Invent. Math., 99 (1990), 205–213.

    Article  MathSciNet  MATH  Google Scholar 

  20. J. Heinonen, Lectures on Analysis on Metric Spaces, Springer, New York, 2001.

    MATH  Google Scholar 

  21. J. Heinonen, The branch set of a quasiregular mapping, in Proceedings of the International Congress of Mathematicians, vol. II (Beijing, 2002), pp. 691–700, Higher Ed. Press, Beijing, 2002.

    Google Scholar 

  22. J. Heinonen, Lectures on Lipschitz Analysis, Report. University of Jyväskylä Department of Mathematics and Statistics, vol. 100, University of Jyväskylä, Jyväskylä, 2005.

    MATH  Google Scholar 

  23. J. Heinonen and T. Kilpeläinen, BLD-mappings in W 2,2 are locally invertible, Math. Ann., 318 (2000), 391–396.

    Article  MathSciNet  MATH  Google Scholar 

  24. J. Heinonen and P. Koskela, Quasiconformal maps in metric spaces with controlled geometry, Acta Math., 181 (1998), 1–61.

    Article  MathSciNet  MATH  Google Scholar 

  25. J. Heinonen and S. Rickman, Geometric branched covers between generalized manifolds, Duke Math. J., 113 (2002), 465–529.

    Article  MathSciNet  MATH  Google Scholar 

  26. J. Heinonen and D. Sullivan, On the locally branched Euclidean metric gauge, Duke Math. J., 114 (2002), 15–41.

    Article  MathSciNet  MATH  Google Scholar 

  27. J. Heinonen, P. Koskela, N. Shanmugalingam, and J. T. Tyson, Sobolev classes of Banach space-valued functions and quasiconformal mappings, J. Anal. Math., 85 (2001), 87–139.

    Article  MathSciNet  MATH  Google Scholar 

  28. M. W. Hirsch and B. Mazur, Smoothings of Piecewise Linear Manifolds, Annals of Mathematics Studies, vol. 80, Princeton University Press, Princeton, 1974.

    MATH  Google Scholar 

  29. S. Keith, Modulus and the Poincaré inequality on metric measure spaces, Math. Z., 245 (2003), 255–292.

    Article  MathSciNet  MATH  Google Scholar 

  30. S. Keith, A differentiable structure for metric measure spaces, Adv. Math., 183 (2004), 271–315.

    Article  MathSciNet  MATH  Google Scholar 

  31. S. Keith, Measurable differentiable structures and the Poincaré inequality, Indiana Univ. Math. J., 53 (2004), 1127–1150.

    Article  MathSciNet  MATH  Google Scholar 

  32. S. Keith and X. Zhong, The Poincaré inequality is an open ended condition, Ann. Math., 167 (2008), 575–599.

    Article  MathSciNet  MATH  Google Scholar 

  33. R. C. Kirby and L. C. Siebenmann, Foundational Essays on Topological Manifolds, Smoothings, and Triangulations, Annals of Mathematics Studies, vol. 80, Princeton University Press, Princeton, 1977. With notes by John Milnor and Michael Atiyah.

    MATH  Google Scholar 

  34. P. Koskela, Upper gradients and Poincaré inequalities, in Lecture Notes on Analysis in Metric Spaces (Trento, 1999), pp. 55–69, Appunti Corsi Tenuti Docenti Sc. Scuola Norm. Sup., Pisa, 2000.

    Google Scholar 

  35. T. J. Laakso, Plane with A ∞-weighted metric not bi-Lipschitz embeddable to \(\Bbb{R}\sp N\), Bull. Lond. Math. Soc., 34 (2002), 667–676.

    Article  MathSciNet  MATH  Google Scholar 

  36. J. Luukkainen and J. Väisälä, Elements of Lipschitz topology, Ann. Acad. Sci. Fenn., Ser. A 1 Math., 3 (1977), 85–122.

    MATH  Google Scholar 

  37. O. Martio and J. Väisälä, Elliptic equations and maps of bounded length distortion, Math. Ann., 282 (1988), 423–443.

    Article  MathSciNet  MATH  Google Scholar 

  38. P. Mattila, Geometry of Sets and Measures in Euclidean Spaces, Cambridge Studies in Advanced Mathematics, vol. 44, Cambridge University Press, Cambridge, 1995.

    MATH  Google Scholar 

  39. J. Milnor, Topological manifolds and smooth manifolds, in Proc. Internat. Congr. Mathematicians (Stockholm, 1962), pp. 132–138, Inst. Mittag-Leffler, Djursholm, 1963.

    Google Scholar 

  40. J. W. Milnor and J. D. Stasheff, Characteristic Classes, Annals of Mathematics Studies, vol. 76, Princeton University Press, Princeton, 1974.

    MATH  Google Scholar 

  41. E. E. Moise, Geometric Topology in Dimensions 2 and 3, Graduate Texts in Mathematics, vol. 47, Springer, New York, 1977.

    MATH  Google Scholar 

  42. S. Müller and V. Šverák, On surfaces of finite total curvature, J. Differ. Geom., 42 (1995), 229–258.

    MATH  Google Scholar 

  43. J. Munkres, Obstructions to imposing differentiable structures, Ill. J. Math., 8 (1964), 361–376.

    MathSciNet  MATH  Google Scholar 

  44. C. C. Pugh, Smoothing a topological manifold, Topol. Appl., 124 (2002), 487–503.

    Article  MathSciNet  MATH  Google Scholar 

  45. F. Quinn, An obstruction to the resolution of homology manifolds, Mich. Math. J., 34 (1987), 285–291.

    Article  MathSciNet  MATH  Google Scholar 

  46. Y. G. Reshetnyak, Space mappings with bounded distortion, Sib. Mat. Zh., 8 (1967), 629–659.

    MATH  Google Scholar 

  47. Y. G. Reshetnyak, Space Mappings with Bounded Distortion, Translations of Mathematical Monographs, vol. 73, American Mathematical Society, Providence, 1989. Translated from the Russian by H. H. McFaden.

    MATH  Google Scholar 

  48. S. Rickman, Quasiregular Mappings, Springer, Berlin, 1993.

    MATH  Google Scholar 

  49. S. Semmes, Bi-Lipschitz mappings and strong A ∞ weights, Ann. Acad. Sci. Fenn., Ser. A 1 Math., 18 (1993), 211–248.

    MathSciNet  Google Scholar 

  50. S. Semmes, Finding curves on general spaces through quantitative topology, with applications to Sobolev and Poincaré inequalities, Sel. Math., 2 (1996), 155–295.

    Article  MathSciNet  MATH  Google Scholar 

  51. S. Semmes, Good metric spaces without good parameterizations, Rev. Mat. Iberoam., 12 (1996), 187–275.

    MathSciNet  MATH  Google Scholar 

  52. S. Semmes, On the nonexistence of bi-Lipschitz parameterizations and geometric problems about A ∞-weights, Rev. Mat. Iberoam., 12 (1996), 337–410.

    MathSciNet  MATH  Google Scholar 

  53. Y. Shikata, On a distance function on the set of differentiable structures, Osaka J. Math., 3 (1966), 65–79.

    MathSciNet  MATH  Google Scholar 

  54. Y. Shikata, On the smoothing problem and the size of a topological manifold, Osaka J. Math., 3 (1966), 293–301.

    MathSciNet  MATH  Google Scholar 

  55. L. Siebenmann and D. Sullivan, On complexes that are Lipschitz manifolds, in Geometric Topology, pp. 503–525, Academic Press, New York, 1979.

    Google Scholar 

  56. D. Sullivan, Hyperbolic geometry and homeomorphisms, in Geometric Topology (Proc. Georgia Topology Conf., Athens, GA., 1977), pp. 543–555, Academic Press, New York, 1979.

    Google Scholar 

  57. D. Sullivan, Exterior d, the local degree, and smoothability, in Prospects in Topology (Princeton, NJ, 1994), Ann. of Math. Stud., vol. 138, pp. 328–338, Princeton Univ. Press, Princeton, 1995.

    Google Scholar 

  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. D. Sullivan, On the Foundation of Geometry, Analysis, and the Differentiable Structure for Manifolds, in Topics in Low-Dimensional Topology (University Park, PA, 1996), pp. 89–92, World Sci. Publishing, River Edge, 1999.

    Google Scholar 

  60. N. Teleman, The index of signature operators on Lipschitz manifolds, Inst. Hautes Études Sci. Publ. Math., 58 (1984), 39–78.

    Google Scholar 

  61. T. Toro, Surfaces with generalized second fundamental form in L 2 are Lipschitz manifolds, J. Differ. Geom., 39 (1994), 65–101.

    MathSciNet  MATH  Google Scholar 

  62. T. Toro, Geometric conditions and existence of bi-Lipschitz parameterizations, Duke Math. J., 77 (1995), 193–227.

    Article  MathSciNet  MATH  Google Scholar 

  63. J. Väisälä, Minimal mappings in Euclidean spaces, Ann. Acad. Sci. Fenn., Ser. A 1 Math., 366 (1965), 22.

    Google Scholar 

  64. J. Väisälä, Exhaustions of John domains, Ann. Acad. Sci. Fenn., Ser. A 1 Math., 19 (1994), 47–57.

    MATH  Google Scholar 

  65. J. H. C. Whitehead, Manifolds with transverse fields in Euclidean space, Ann. Math., 73 (1961), 154–212.

    Article  MathSciNet  Google Scholar 

  66. H. Whitney, Geometric Integration Theory, Princeton University Press, Princeton, 1957.

    MATH  Google Scholar 

  67. K. Wildrick, Quasisymmetric structures on surfaces. Preprint, September 2007.

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

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Authors and Affiliations

  1. Department of Mathematics, University of Michigan, Ann Arbor, MI, 48109, USA

    Juha Heinonen

  2. Sydney, Australia

    Stephen Keith

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  1. Juha Heinonen
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  2. Stephen Keith
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Correspondence to Stephen Keith.

Additional information

Juha Heinonen passed away during the very final preparation of this paper. S.K. dedicates this work to his memory.

The first author was supported by the NSF grants DMS 0244421, 0353549, and 0652915.

The second author (Stephen Keith) was employed at the University of Helsinki and at the Centre for Mathematics and its Application at Australia National University during part of the research for this paper.

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Heinonen, J., Keith, S. Flat forms, bi-Lipschitz parametrizations, and smoothability of manifolds. Publ.math.IHES 113, 1–37 (2011). https://doi.org/10.1007/s10240-011-0032-4

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