Inventiones mathematicae

, Volume 120, Issue 1, pp 61–79 | Cite as

Definitions of quasiconformality

  • Juha Heinonen
  • Pekka Koskela


We establish that the infinitesimal “H-definition” for quasiconformal mappings on Carnot groups implies global quasisymmetry, and hence the absolute continuity on almost all lines. Our method is new even inR n where we obtain that the “limsup” condition in theH-definition can be replaced by a “liminf” condition. This leads to a new removability result for (quasi)conformal mappings in Euclidean spaces. An application to parametrizations of chord-arc surfaces is also given.


Euclidean Space Conformal Mapping Quasiconformal Mapping Absolute Continuity Carnot Group 
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  1. [AK]
    Astala K., Koskela P.: Quasiconformal mappings and global integrability of the derivative. J. Anal. Math.57, 203–220 (1991)Google Scholar
  2. [B]
    Bojarski B.: Remarks on Sobolev imbedding inequalities, In Proc. of the Conference on Complex Analysis, Joensuu 1987. Lecture Notes in Math. 1351, Springer Verlag 1988Google Scholar
  3. [C]
    Coornaert M.: Mesures de Patterson-Sullivan sur le bord d'un espace hyperbolique au sens de Gromov, Pacific J. Math.159, 241–270 (1993)Google Scholar
  4. [DS]
    David G., Semmes S.: Quantitative rectifiability and Lipschitz mappings. Trans. Amer. Math. Soc.337, 855–889 (1993)Google Scholar
  5. [E]
    Eichmann R.: Variationsprobleme auf der Heisenberggruppe, Lizentiatsarbeit. Universität Bern (1990)Google Scholar
  6. [F]
    Federer H.: Geometric Measure Theory, Springer, New York, 1969.Google Scholar
  7. [FS]
    Folland G.B., Stein E.M.: Hardy spaces on homogeneous groups, Princeton University Press, Princeton, New Jersey, 1982Google Scholar
  8. [G1]
    Gehring F.W.: The definitions and exceptional sets for quasiconformal mappings. Ann. Acad. Sci. Fenn. Ser. A I Math.281, 1–28 (1960)Google Scholar
  9. [G2]
    Gehring F.W.: Extremal length definitions for the conformal capacity of rings in space, Michigan Math. J.9, 137–150 (1962)Google Scholar
  10. [GH]
    Ghys E., de la Harpe P.: Sur les Groupes Hyperboliques d'après Mikhaer Gromov, Birkhäuser, Progress in Mathematics, Boston-Basel-Berlin, 1990Google Scholar
  11. [GP]
    Gromov M., Pansu P.: Rigidity of Lattices: An Introduction, Geometric Topology: Recent Developments. Lecture Notes in Mathematics 1504, Springer-Verlag. Berlin-New York-Heidelberg, 1991Google Scholar
  12. [HS]
    He Z.-X., Schramm O.: Rigidity of circle domains whose boundary has σ-finite linear measure, Invent. Math.115, 297–310 (1994)Google Scholar
  13. [H1]
    Heinonen J.: A capacity estimate on Carnot groups, Bull. Sci. Math. Fr. (to appear)Google Scholar
  14. [H2]
    Heintze E.: On homogeneous manifolds of negative curvature, Math. Ann.211, 23–34 (1974)Google Scholar
  15. [KR]
    Korányi A., Reimann H.M.: Foundations for the theory of quasiconformal mappings on the Heisenberg group, Adv. in Math. (to appear)Google Scholar
  16. [L]
    Loewner C.: On the conformal capacity in space. J. Math. Mech.8, 411–414 (1959)Google Scholar
  17. [MN]
    Martio O., Näkki R.: Continuation of quasiconformal mappings, (in Russian), Sib. Mat. Zh.28, 162–170 (1987), English translation: Siberian Math. J.28, 645–652 (1988)Google Scholar
  18. [M1]
    Mattila P.: Geometry of sets and measures in Euclidean spaces, to appear in Cambridge Univ. PressGoogle Scholar
  19. [M2]
    Mitchell J.: On Carnot-Carathéodory metrices, J. Diff. Geom.21, 35–45 (1985)Google Scholar
  20. [M3]
    Mostow G.D.: Strong rigidity of locally symmetric spaces, Princeton University Press, Princeton, New Jersey, 1973Google Scholar
  21. [M4]
    Mostow G.D.: A remark on quasiconformal mappings on Carnot groups. Michigan Math. J.41, 31–37 (1994)Google Scholar
  22. [P1]
    Pansu P.: Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un, Ann. Math.129, 1–60 (1989)Google Scholar
  23. [P2]
    Pansu P.: Dimension conforme et sphère à l'infini des veriétés à courbure négative, Ann. Acad. Sci. Fenn. Ser. A I Math.,14, 177–212 (1989)Google Scholar
  24. [P3]
    Paulin F.: Un groupe hyperbolique est déterminé par son bord, preprint (1993)Google Scholar
  25. [R]
    Reimann H.M.: An estimate for pseudoconformal capacities on the sphere, Ann. Acad. Sci. Fenn. Ser. A I Math14, 315–324 (1989)Google Scholar
  26. [S]
    Semmes S.: Chord-arc surfaces with small constant. II. Good parameterizations, Adv. in Math.88, 170–199 (1989)Google Scholar
  27. [TV]
    Tukia P., Väisälä J.: Quasisymmetric embeddings of metric spaces, Ann. Acad. Sci. Fenn. Ser. A I Math.5, 97–114 (1980)Google Scholar
  28. [V1]
    Väisälä J. Lectures on n-dimensional quasiconformal mappings, Lecture Notes in Math.229, Springer-Verlag, Berlin-Heidelberg-New York, 1971Google Scholar
  29. [V2]
    Väisälä J.: Quasisymmetric embeddings in euclidean spaces, Trans. Amer. Math. Soc.264, 191–204 (1981)Google Scholar
  30. [V3]
    Väisälä J. Quasisymmetric maps of products of curves into the plane, Rev. Roumaine Math. Pures Appl.33, 147–156 (1988)Google Scholar
  31. [Z]
    Ziemer W.P.: Extremal length and p-capacity, Michigan Math. J.16, 43–51 (1969)Google Scholar

Copyright information

© Springer-Verlag 1995

Authors and Affiliations

  • Juha Heinonen
    • 1
  • Pekka Koskela
    • 1
  1. 1.Department of MathematicsUniversity of MichiganAnn ArborUSA

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