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On the Asymptotic Behavior of Quasiconformal Mappings in Space

  • V. Ya. Gutlyanskiǐ
  • O. Martio
  • V. I. Ryazanov
  • M. Vuorinen

Abstract

Conformality at a point of a space quasiconformal mapping is studied without the assumption of differentiability. First, we introduce a notion of asymptotic linearity for general mappings at a prescribed point. Then we prove that the simultaneous asymptotic linearity and analyticity of a mapping f at a point of discreteness imply that f preserves the angles between rays emanating from this point and the moduli of infinitesimal annuli centered at it and thus we strengthen an earlier result of Caraman. Finally, we give sufficient conditions for quasiconformal mappings to be asymptotically linear and analytic in terms of the distortion coefficient and, as a consequence, we generalize the well-known Belinskii weak conformality result to the n-dimensional case.

Keywords

Quasiconformal Mapping Quasiregular Mapping Asymptotic Linearity Distortion Coefficient Prescribe Point 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [A]
    Agard S., Angles and quasiconformal mappings in space,J. Analyse Math., 22 (1969), 177–200.MathSciNetMATHGoogle Scholar
  2. [AG]
    Agard S., Gehring F.W., Angles and quasiconformal mappings, Proc. London Math. Soc. 14A (1965), no.3, 1–21.MathSciNetCrossRefGoogle Scholar
  3. [ABL]
    Anderson J.M., Becker J, Lesley F.D., On the boundary correspondence of asymptotically conformal automorphisms, J. London Math. Soc., 38 (1988), no.3, 453–462.MathSciNetMATHGoogle Scholar
  4. [AVV1]
    Anderson G.D., Vamanamurthy M.K., Vuorinen M., Conformal invariants, quasiconformal maps and special functions, Lecture Notes Math., 1508 (1992), 1–19.MathSciNetCrossRefGoogle Scholar
  5. [AVV2]
    Anderson G.D., Vamanamurthy M.K., Vuorinen M., Dimension-free quasiconformal distortion in n-space, Trans. Amer. Math. Soc., 297 (1986), no.2, 687–706.MathSciNetMATHGoogle Scholar
  6. [BP]
    Becker J., Pommerenke Ch., Uber die quasikonforme Fortsetzung schlichten Funktionen, Math. Zeit., 161 (1978), 69–80.MathSciNetMATHCrossRefGoogle Scholar
  7. [B]
    Belinskii P.P., General properties of quasiconformal mappings, Nauka, Novosibirsk, 1974, Russian.Google Scholar
  8. [BI]
    Bojarski B., Iwaniec T., Another approach to Liouville theorem, Math. Nachr.,107 (1982), 253–262.MathSciNetMATHCrossRefGoogle Scholar
  9. [BJ]
    Brakalova M., Jenkins J A, On the local behavior of certain homeomorphisms, Kodai Math. J., 17 (1994), 201–213.MathSciNetMATHCrossRefGoogle Scholar
  10. [C]
    Caraman P., n-dimensional quasicon formal mappings, Bucuresti-Kent, 1974.Google Scholar
  11. [CA]
    Carleson L., On mappings conformal at the boundary, J. Analyse Math., 19 (1967), 1–13.MathSciNetMATHCrossRefGoogle Scholar
  12. [DS]
    Donaldson S., Sullivan D., Quasiconformal 4-manifolds, Acta Math.,163 (1989), 181–252.MathSciNetMATHCrossRefGoogle Scholar
  13. [D]
    Drasin D., On the Teichmüller-Wittich-Belinskii theorem, Results in Mathematics, 10 (1986), 54–65.MathSciNetMATHGoogle Scholar
  14. [DU]
    Dugundji J., Topology, Allyn Bacon, Inc., Boston, 1974.Google Scholar
  15. [F]
    Fehlmann R., Uber extremale quasikonforme Abbildungen, Comment. Math. Helv., 56 (1981), 558–580.MathSciNetMATHCrossRefGoogle Scholar
  16. [Fer]
    Ferrand J., Regularity of conformal mappings of Riemannian manifolds, Lect. Notes in Math., 743 (1979), 197–203.MathSciNetCrossRefGoogle Scholar
  17. [GS]
    Gardiner F.P., Sullivan D.P., Symmetric structures on a closed curve, Amer. J. Math.,114 (1992), 683–736.MathSciNetMATHCrossRefGoogle Scholar
  18. [G1]
    Gehring F.W., Rings and quasiconformal mappings in space, Trans. Amer. Math. Soc., 103 (1962), 353–393.MathSciNetMATHCrossRefGoogle Scholar
  19. [G2]
    Gehring F.W., Quasiconformal mappings, in Complex analysis and its applications, 2, International Atomic Energy Agency, Vienna, 1976, pp. 213–268.Google Scholar
  20. [GL]
    Gehring F.W., Lehto O., On the total differentiability of functions of a complex variable, Ann. Acad. Sci. Fenn. Ser. A.I., 272 (1960), 1–9.Google Scholar
  21. [GR1]
    Gutlyanskii, V.Ya., Ryazanov, V.I., On asymptotically conformal curves, Complex Variables J., 25 (1994), no.4, 357–366.MathSciNetCrossRefGoogle Scholar
  22. [GR2]
    Gutlyanskii, V.Ya., Ryazanov, V.I., On boundary correspondence under quasiconformal mappings, Ann. Acad. Sci. Fenn. Ser. Math., 21 (1996), 167–178.MathSciNetGoogle Scholar
  23. [H]
    Halmos P.R., Finite-dimensional vector spaces, 2nd. ed., Van Nostrand Reinhold, New York, etc., 1958.Google Scholar
  24. [HKM]
    Heinonen J., Kilpelinen, T., Martio, O., Nonlinear Potential Theory of Degenerate Elliptic Equations,Clarendon Press, Oxford-New York-Tokyo, 1993.Google Scholar
  25. [Iw]
    Iwaniec T., Regularity theorems for solutions of partial differential equations related to quasiregular mappings in several variables, Dissertationes Mathematicae, CXCVIII (1982), 1–48.Google Scholar
  26. [IM]
    Iwaniec T., Martin G., Quasiregular mappings in even dimensions, Acta Math., 170 (1993), 29–81.MathSciNetMATHGoogle Scholar
  27. [K]
    Kudryavtseva N.A., On a condition of weak conformality, Sibirskii Mat. Zh., 35 (1994), no.2, 377–379.MathSciNetGoogle Scholar
  28. [L]
    Lehto O., On the differentiability of quasiconformal mappings with prescribed complex dilatation, Ann. Acad. Sci. Fenn. A I, 275 (1960), 1–28.MathSciNetGoogle Scholar
  29. [LV]
    Lehto O., Virtanen K., Quasikonforme Abbildungen, Springer-Verlag, Berlin, etc., 1965.Google Scholar
  30. [MRV]
    Martio O., Rickman S., Väisälä J., Distortion and singularities of quasiregular mappings, Ann. Acad. Sci. Fenn. Ser. Al, 465 (1970), 1–13.Google Scholar
  31. [M]
    Mostow G.D., Quasiconformal mappings in n-space and rigidity of hyperbolic space forms, Inst. Hautes Etudes Sci. Publ. Math.,34 (1968), 53–104.MathSciNetMATHCrossRefGoogle Scholar
  32. [Rel]
    Reshetnyak Yu.G., Stability of conformal mappings in multi-dimensional spaces, Sibir. Mat. Zh., 8 (1967), 91–114.MATHGoogle Scholar
  33. [Re2]
    Reshetnyak Yu.G., Stability theorems in geometry and analysis, Nauka, Sibirskoe otdelenie, Novosibirsk, 1982, Russian.Google Scholar
  34. [Re3]
    Reshetnyak Yu.G, Space Mappings with Bounded Distortion, Transl. of Math. Monographs, AMS, 73 (1989).Google Scholar
  35. [RW]
    Reich E., Walczak R.K., On the behavior of quasiconformal mappings at a point, Trans. Amer. Math. Soc., 117 (1965), 338–351.MathSciNetMATHCrossRefGoogle Scholar
  36. [S]
    Saks S., Theory of the integral, Dover Publ. Inc., New York, 1937.Google Scholar
  37. [Sch]
    Schatz A., On the local behavior of homeomorphic solutions of Beltrami’s equations, Duke Math. J., 35 (1968), 289–306.MathSciNetMATHCrossRefGoogle Scholar
  38. [St]
    Strebel K., Ein Konvergenzsatz für Folgen quasikonformen Abbildungen, Comment. Math. Helv., 44 (1969), 469–475.MathSciNetMATHCrossRefGoogle Scholar
  39. [T]
    Teichmüller O., Untersuchungen über konforme und quasikonforme Abbildung, Deutsche Math.,3 (1938), 621–678.Google Scholar
  40. [TV]
    Tukia P., Väisälä J., A remark on 1-quasiconformal maps, Ann. Acad. Sci. Fenn. Ser. A, 10 (1985), 561–562.MATHGoogle Scholar
  41. [V1]
    Väisälä J., On quasiconformal mappings in space, Ann. Acad. Sci. Fenn. Ser. A, 298 (1961), 1–36.Google Scholar
  42. [V2]
    Väisälä J., Lectures on n-Dimensional Quasiconformal Mappings, Lecture Notes in Math., 229, Springer-Verlag, Berlin-New York, 1971.MATHGoogle Scholar
  43. [Vul]
    Vuorinen M., Conformal Geometry and Quasiregular Mappings, Lecture Notes in Math., 1319, Springer-Verlag, Berlin-New York, 1988.Google Scholar
  44. [Vu2]
    Vuorinen M., Quadruples and spatial quasiconformal mappings, Math. Z.,205 (1990), 617–628.MathSciNetMATHCrossRefGoogle Scholar
  45. [W]
    Wittich H., Zum Beweis eines Satzes über quasikonforme Abbildungen, Math. Z.,51 (1948), 275–288.MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1998

Authors and Affiliations

  • V. Ya. Gutlyanskiǐ
    • 1
    • 2
  • O. Martio
    • 1
    • 2
  • V. I. Ryazanov
    • 1
    • 2
  • M. Vuorinen
    • 1
    • 2
  1. 1.Institute of Applied Mathematics and MechanicsNAS of UkraineDonetskUkraine
  2. 2.Department of MathematicsUniversity of HelsinkiHelsinkiFinland

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