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Journal of Geometric Analysis

, Volume 20, Issue 3, pp 651–669 | Cite as

A Conformal de Rham Complex

Article

Abstract

We introduce the notion of a conformal de Rham complex of a Riemannian manifold. This is a graded differential Banach algebra and it is invariant under quasiconformal maps, in particular the associated cohomology is a new quasiconformal invariant.

Keywords

Quasiconformal maps Conformal invariants Lq,p-cohomology 

Mathematics Subject Classification (2000)

30C65 58A10 58A12 53c 

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References

  1. 1.
    Bers, L.: Mathematical Aspects of Subsonic and Transonic Gas Dynamics. Wiley, New York (1958) MATHGoogle Scholar
  2. 2.
    Burenkov, V.: Sobolev Spaces on Domains. Teubner-Texte zur Mathematik (1998) Google Scholar
  3. 3.
    Donaldson, S., Sullivan, D.: Quasiconformal 4-manifolds. Acta Math. 163, 181–252 (1989) MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Gilbarg, D., Trudinger, N.: Elliptic Partial Differential Equations of Second Order, 2nd edn. Grundlehren der Mathematischen Wissenschaften, vol. 224. Springer, Berlin (1983) MATHGoogle Scholar
  5. 5.
    Gol’dshtein, V.M., Reshetnyak, Y.G.: Quasiconformal Mappings and Sobolev Spaces. Kluwer Academic, Dordrecht (1990) MATHGoogle Scholar
  6. 6.
    Gol’dshtein, V., Troyanov, M.: The L pq-cohomology of SOL. Annales de la Faculte des Sciences de Toulouse, vol. Vii, no. 4 (1998) Google Scholar
  7. 7.
    Gol’dshtein, V., Troyanov, M.: The Kelvin-Nevanlinna-Royden criterion for p-parabolicity. Math. Z. 232, 607–619 (1999) MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Gol’dshtein, V., Troyanov, M.: Sobolev inequality for differential forms and L qp-cohomology. J. Geom. Anal. 16(4), 597–631 (2006) MATHMathSciNetGoogle Scholar
  9. 9.
    Gol’dshtein, V., Troyanov, M.: On the naturality of the exterior differential. C. R. Math. Acad. Sci. Soc. R. Can. 30(1), 1–10 (2008) MATHMathSciNetGoogle Scholar
  10. 10.
    Gromov, M.: Asymptotic invariants of infinite groups. In: Niblo, G., Roller, M. (eds.) Geometric Group Theory. Cambridge University Press, Cambridge (1993) Google Scholar
  11. 11.
    Heinonen, J.: What is … a quasiconformal mapping? Not. Am. Math. Soc. 53(11), 1334–1335 (2006) MATHMathSciNetGoogle Scholar
  12. 12.
    Heinonen, J., Kilpeläinen, T., Martio, O.: Non-Linear Potential Theory of Degenerate Elliptic Equations. Oxford Math. Monographs (1993) Google Scholar
  13. 13.
    Holopainen, I.: Non-linear potential theory and quasiregular mappings on riemannian manifolds. Ann. Acad. Sci. Fennicae, Ser. A 74 (1990) Google Scholar
  14. 14.
    Lelong-Ferrand, J.: Etude d’une classe d’applications liées à des homomorphismes d’algèbres de fonctions et généralisant les quasi-conformes. Duke Math. J. 40, 163–186 (1973) MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Pansu, P.: Cohomologie L p et pincement. Comment. Math. Helv. 83(2), 327–357 (2008) MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Reimann, M.: Functions of bounded mean oscillation and quasiconformal mappings. Comment. Math. Helv. 49, 260–276 (1974) MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Reshetnyak, Y.G.: Space Mappings with Bounded Distortion. Translations of Mathematical Monographs, vol. 73. AMS, Providence (1989) MATHGoogle Scholar
  18. 18.
    Rickman, S.: Quasiregular Mappings. Springer, Berlin (1993) MATHGoogle Scholar
  19. 19.
    Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton Mathematical Series, vol. 30. Princeton University Press, Princeton (1970) MATHGoogle Scholar
  20. 20.
    Sullivan, D.: In: Cantrell, J. (ed.) Hyperbolic Geometry and Homeomorphisms, Proc. Georgia Conf. on Geometric Topology. Academic Press, New York (1978) Google Scholar
  21. 21.
    Troyanov, M.: L’Horizon de SOL. Exp. Math. 16, 441–479 (1998) MATHMathSciNetGoogle Scholar
  22. 22.
    Troyanov, M.: Parabolicity of manifolds. Sib. Adv. Math. 9, 125–150 (1999) MATHMathSciNetGoogle Scholar
  23. 23.
    Väisälä, J.: Lectures on n-dimensional Quasiconformal Mappings. Lect. Notes in Math., vol. 229. Springer, Berlin (1971) Google Scholar
  24. 24.
    Vodop’yanov, S.K., Gol’dshtein, V.M.: Lattice isomorphisms of the spaces \(W^{1}_{n}\) and quasiconformal mappings. Sib. Math. J. 16(2), 174–189 (1975) MATHCrossRefGoogle Scholar
  25. 25.
    Vuorinen, M.: Quasiconformal Space Mappings—A Collection of Surveys 1960–1990. Springer Lect. Notes in Math., vol. 1508. Springer, Berlin (1992) MATHCrossRefGoogle Scholar
  26. 26.
    Vuorinen, M.: Geometric properties of quasiconformal maps. arXiv:math/0703687v1
  27. 27.
    Zorich, V.A., Kesel’man, V.M.: On the conformal type of a Riemannian manifold. Func. Anal. Appl. 30, 106–117 (1996) MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Mathematica Josephina, Inc. 2010

Authors and Affiliations

  1. 1.Department of MathematicsBen Gurion University of the NegevBeer-ShevaIsrael
  2. 2.Section de MathématiquesÉcole Polytechnique Féderale de LausanneLausanneSwitzerland

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