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Quasiconformal Space Mappings pp 20–38Cite as

Topics in quasiconformal mappings

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References

  1. L. V. Ahlfors, Zur theorie der Überlagerungsflächen, Acta Math. 65 (1935), 157–194.

    Article  MathSciNet  MATH  Google Scholar 

  2. _____, On quasiconformal mappings, J. Analyse Math. 3 (1953/54), 1–58.

    Article  MathSciNet  MATH  Google Scholar 

  3. _____, Quasiconformal reflections, Acta Math. 109 (1963), 291–301.

    Article  MathSciNet  MATH  Google Scholar 

  4. _____, Extension of quasiconformal mappings from two to three dimensions, Proc. Nat. Acad. Sci. U.S.A. 51 (1964), 768–771.

    Article  MathSciNet  MATH  Google Scholar 

  5. _____, Finitely generated Kleinian groups, Amer. J. Math. 86 (1964), 413–429.

    Article  MathSciNet  MATH  Google Scholar 

  6. _____, Quasiconformal mappings, Teichmüller spaces, and Kleinian groups, Proc. Internat. Congr. Math. (Helsinki, 1978), Acad. Sci. Fennica, Helsinki, 1980, pp. 71–84.

    MATH  Google Scholar 

  7. L. V. Ahlfors and L. Bers, Riemann’s mapping theorem for variable metrics, Ann. of Math. 72 (1960), 385–404.

    Article  MathSciNet  MATH  Google Scholar 

  8. L. V. Ahlfors and A. Beurling, Conformal invariants and function-theoretic null sets, Acta Math. 83 (1950), 101–129.

    Article  MathSciNet  MATH  Google Scholar 

  9. K. Astala, A remark on quasi-conformal mappings and BMO-functions, Michigan Math. J. 30 (1983), 209–212.

    Article  MathSciNet  MATH  Google Scholar 

  10. A. Baernstein II and J. J. Manfredi, Topics in quasiconformal mapping, Topics in Modern Harmonic Analysis, Istituto Nazionale di Alta Mathematica, Roma, 1983, pp. 819–862.

    MATH  Google Scholar 

  11. L. Bers, Quasiconformal mappings and Teichmüller’s theorem, Analytic Functions, Princeton Univ. Press, Princeton, N.J., 1960, pp. 89–119.

    MATH  Google Scholar 

  12. _____, Uniformization by Beltrami equations, Comm. Pure Appl. Math. 14 (1961), 215–228.

    Article  MathSciNet  MATH  Google Scholar 

  13. _____, Uniformization, moduli, and Kleinian groups, Bull. London Math. Soc. 4 (1972), 257–300.

    Article  MathSciNet  MATH  Google Scholar 

  14. _____, Quasiconformal mappings, with applications to differential equations, function theory and topology, Bull. Amer. Math. Soc. 83 (1977), 1083–1100.

    Article  MathSciNet  MATH  Google Scholar 

  15. _____, An extremal problem for quasiconformal mappings and a problem of Thurston, Acta Math. 141 (1978), 73–98.

    Article  MathSciNet  MATH  Google Scholar 

  16. _____, Finite dimensional Teichmüller spaces and generalizations, Bull. Amer. Math. Soc. 5 (1981), 131–172.

    Article  MathSciNet  MATH  Google Scholar 

  17. A. Beurling and L. V. Ahlfors, The boundary correspondence under quasiconformal mappings, Acta Math. 96 (1956), 125–142.

    Article  MathSciNet  MATH  Google Scholar 

  18. B. Bojarski, Generalized solutions of a system of first order differential equations of elliptic type with discontinuous coefficients, Mat. Sb. 43 (1957), 451–503. (Russian)

    MathSciNet  Google Scholar 

  19. M. Brown, Locally flat embeddings of topological manifolds, Ann. Math. 75 (1962), 331–341.

    Article  MATH  Google Scholar 

  20. L. Carleson, The extension problem for quasiconformal mappings, Contributions to Analysis, Academic Press, New York, 1974, pp. 39–47.

    MATH  Google Scholar 

  21. A. Douady and C. J. Earle, Conformally natural extension of homeomorphisms of the circle, Acta Math. (to appear).

    Google Scholar 

  22. D. Drasin, The inverse problem of the Nevanlinna theory, Acta Math. 138 (1977), 83–151.

    Article  MathSciNet  MATH  Google Scholar 

  23. C. J. Earle and J. Eells, A fibre bundle description of Teichmüller theory, J. Differential Geom. 3 (1969), 19–43.

    MathSciNet  MATH  Google Scholar 

  24. C. J. Earle and S. Nag, Conformally natural reflections in Jordan curves with applications to Teichmüller spaces (to appear).

    Google Scholar 

  25. T. Erkama, Group actions and extension problems for maps of balls, Ann. Acad. Sci. Fenn. Ser. A I Math. 556 (1973), 1–31.

    MathSciNet  MATH  Google Scholar 

  26. C. Fefferman, Characterizations of bounded mean oscillation, Bull. Amer. Math. Soc. 77 (1971), 587–588.

    Article  MathSciNet  MATH  Google Scholar 

  27. M. H. Freedman and R. Skora, Strange actions of groups on spheres, J. Differential Geom. (to appear).

    Google Scholar 

  28. B. Fuglede, Extremal length and functional completion, Acta Math. 98 (1957), 171–219.

    Article  MathSciNet  MATH  Google Scholar 

  29. D. B. Gauld and J. Väisälä, Lipschitz and quasiconformal flattening of spheres and cells, Ann. Acad. Sci. Fenn. Ser. A I Math. 4 (1978/79), 371–382.

    Article  MathSciNet  MATH  Google Scholar 

  30. F. W. Gehring, Symmetrization of rings in space, Trans. Amer. Math. Soc. 101 (1961), 499–519.

    Article  MathSciNet  MATH  Google Scholar 

  31. ___, Rings and quasiconformal mappings in space, Trans. Amer. Math. Soc. 103 (1962), 353–393.

    Article  MathSciNet  MATH  Google Scholar 

  32. ___, Extension of quasiconformal mappings in three space, J. Analyse Math. 14 (1965), 171–182.

    Article  MathSciNet  MATH  Google Scholar 

  33. ___, Extension theorems for quasiconformal mappings in n-space, J. Analyse Math. 19 (1967), 149–169.

    Article  MathSciNet  MATH  Google Scholar 

  34. ___, The L p-integrability of the partial derivatives of a quasiconformal mapping, Acta Math. 130 (1973), 265–277.

    Article  MathSciNet  MATH  Google Scholar 

  35. ___, Quasiconformal mappings, Complex Analysis and its Applications. II, International Atomic Energy Agency, Vienna, 1976, pp. 213–268.

    Google Scholar 

  36. ___, Injectivity of local quasi-isometries, Comment. Math. Helv. 57 (1982), 202–220.

    Article  MathSciNet  MATH  Google Scholar 

  37. ___, Extension of quasiisometric embeddings of Jordan curves, Complex Variables Theory Appl. 5 (1986), 245–263.

    Article  MathSciNet  MATH  Google Scholar 

  38. F. W. Gehring and G. J. Martin, Discrete quasiconformal groups I, Proc. London Math. Soc. (to appear).

    Google Scholar 

  39. F. W. Gehring and E. Reich, Area distortion under quasiconformal mappings, Ann. Acad. Sci. Fenn Ser. A I Math. 388 (1966), 1–15.

    MathSciNet  MATH  Google Scholar 

  40. F. W. Gehring and J. Väisälä, The coefficients of quasiconformality of domains in space, Acta Math. 114 (1965), 1–70.

    Article  MathSciNet  MATH  Google Scholar 

  41. M. Giaquinta, Multiple integrals in the calculus of variations and nonlinear elliptic systems, Ann. of Math. Studies, Vol. 105, Princeton Univ. Press, Princeton, N.J., 1983.

    MATH  Google Scholar 

  42. H. Grötzsch, Über möglichst konforme Abbildungen von schlichten Bereichen, Ber. Verh. Sächs. Akad. Wiss. Leipzig 84 (1932), 114–120.

    MATH  Google Scholar 

  43. T. Iwaniec, Extremal inequalities in Sobolev spaces and quasiconformal mappings, Z. Anal. Anwendungen 1 (1982), 1–16.

    MathSciNet  MATH  Google Scholar 

  44. ___, On L p-integrability in PDE’s and quasiregular mappings for large exponents, Ann. Acad. Sci. Fenn. Ser. A I Math. 7 (1982), 301–322.

    Article  MathSciNet  MATH  Google Scholar 

  45. F. John, Rotation and strain, Comm. Pure Appl. Math. 14 (1961), 391–413.

    Article  MathSciNet  MATH  Google Scholar 

  46. ___, On quasi-isometric mappings, II, Comm. Pure Appl. Math. 22 (1969), 265–278.

    Article  MathSciNet  MATH  Google Scholar 

  47. F. John and L. Nirenberg, On functions of bounded mean oscillation, Comm. Pure Appl. Math. 14 (1961), 415–426.

    Article  MathSciNet  MATH  Google Scholar 

  48. P. W. Jones, Extension theorems for BMO, Indiana Univ. Math. J. 29 (1980), 41–66.

    Article  MathSciNet  MATH  Google Scholar 

  49. M. Kiikka, Diffeomorphic approximation of quasiconformal and quasisymmetric homeomorphisms, Ann. Acad. Fenn. Sci. Ser. A I Math. 8 (1983), 251–256.

    Article  MathSciNet  MATH  Google Scholar 

  50. I. Kra, On the Nielsen-Thurston-Bers type of some self-maps of Riemann surfaces, Acta Math. 146 (1981), 231–270.

    Article  MathSciNet  MATH  Google Scholar 

  51. T. Kuusalo, Quasiconformal mappings without boundary extensions, Ann. Acad. Sci. Fenn. Ser. A I Math. 10 (1985), 331–338.

    Article  MathSciNet  MATH  Google Scholar 

  52. M. A. Lavrentieff, Sur une classe de représentations continues, Math. Sb. 42 (1935), 407–423.

    MATH  Google Scholar 

  53. ___, Sur un critère différentiel des transformations homéomorphes des domaines à trois dimensions, Dokl. Akad. Nauk. 20 (1938), 241–242.

    MATH  Google Scholar 

  54. O. Lehto and K. I. Virtanen, Quasiconformal mappings in the plane, Springer-Verlag, 1973.

    Google Scholar 

  55. J. Lelong-Ferrand, Transformations conformes et quasi-conformes des variétés riemanniennes compactes (Démonstration de la conjoncture de A. Lichnerowicz), Acad. Roy. Belg. Cl. Sci. Mém. Collect. 39 (1971), 1–44.

    MathSciNet  MATH  Google Scholar 

  56. , Etude d’une classe d’applications liées à des homomorphismes d’algèbres de fonctions, et généralisant les quasi conformes, Duke Math. J. 40 (1973), 163–186.

    Article  MathSciNet  MATH  Google Scholar 

  57. L. G. Lewis, Quasiconformal mappings and Royden algebras in space, Trans. Amer. Math. Soc. 158 (1971), 481–492.

    Article  MathSciNet  MATH  Google Scholar 

  58. C. Loewner, On the conformal capacity in space, J. Math. Mech. 8 (1959), 411–414.

    MathSciNet  MATH  Google Scholar 

  59. O. Martio, S. Rickman, and J. Väisälä, Topological and metric properties of quasiregular mappings, Ann. Acad. Sci. Fenn. Ser. A I Math. 488 (1971), 1–31.

    MATH  Google Scholar 

  60. B. Mazur, On embeddings of spheres, Bull. Amer. Math. Soc. 65 (1959), 59–65.

    Article  MathSciNet  MATH  Google Scholar 

  61. D. Menchoff, Sur une généralisation d’un théorème de M. H. Bohr, Mat. Sb. 44 (1937), 339–354.

    MATH  Google Scholar 

  62. C. B. Morrey, On the solutions of quasi-linear elliptic partial differential equations, Trans. Amer. Math. Soc. 43 (1938), 126–166.

    Article  MathSciNet  MATH  Google Scholar 

  63. G. D. Mostow, Quasi-conformal mappings in n-space and the rigidity of hyperbolic space forms, Inst. Hautes Études Sci. Publ. Math. 34 (1968), 53–104.

    Article  MathSciNet  MATH  Google Scholar 

  64. Z. Nehari, The Schwarzian derivative and schlicht functions, Bull. Amer. Math. Soc. 55 (1949), 545–551.

    Article  MathSciNet  MATH  Google Scholar 

  65. R. Nirenberg, On quasi-pseudoconformality in several complex variables, Trans. Amer. Math. Soc. 127 (1967), 233–240.

    Article  MathSciNet  MATH  Google Scholar 

  66. H. M. Reimann, Functions of bounded mean oscillation and quasiconformal mappings, Comment. Math. Helv. 49 (1974), 260–276.

    Article  MathSciNet  MATH  Google Scholar 

  67. Yu. G. Reshetnyak, Space mappings with bounded distortion, Sibirsk. Math. Zh. 8 (1967), 629–658. (Russian)

    Google Scholar 

  68. ___, Liouville’s theorem on conformal mappings under minimal regularity assumptions, Sibirsk. Mat. Zh. 8 (1967), 835–840. (Russian)

    Google Scholar 

  69. ___, Stability estimates in Liouville’s theorem and the L p-integrability of the derivatives of quasiconformal mappings, Sibirsk. Mat. Zh. 17 (1976), 868–896. (Russian)

    Google Scholar 

  70. S. Rickman, Extension over quasiconformally equivalent curves, Ann. Acad. Sci. Fenn Ser. A I Math. 436 (1969), 1–12.

    MATH  Google Scholar 

  71. ___, On the number of omitted values of entire quasiregular mappings, J. Analyse Math. 37 (1980), 100–117.

    Article  MathSciNet  MATH  Google Scholar 

  72. D. Sullivan, Hyperbolic geometry and homeomorphisms, Geometric Topology, Academic Press, New York, 1979, pp. 543–555.

    MATH  Google Scholar 

  73. ___, On the ergodic theory at infinity of an arbitrary discrete group of hyperbolic motions, Riemann Surfaces and Related Topics: Proceedings of the 1978 Stony Brook Conference, Ann. of Math. Studies, No. 97, Princeton Univ. Press, Princeton, N.J., 1981, pp. 465–496.

    Google Scholar 

  74. ___, Quasiconformal homeomorphisms and dynamics I. Solution of the Fatou-Julia problem on wandering domains, Ann. of Math. 122 (1985), 401–418.

    Article  MathSciNet  MATH  Google Scholar 

  75. ___, Quasiconformal homeomorphisms and dynamics II: Structural stability implies hyperbolicity for Kleinian groups, Acta Math. 155 (1985), 243–260.

    Article  MathSciNet  MATH  Google Scholar 

  76. O. Teichmüller, Extremale quasikonforme Abbildungen und quadratische Differentiale, Abh. Preuss. Akad. Wiss. Mat.-Nat. Kl. 22 (1940), 1–197.

    MathSciNet  Google Scholar 

  77. P. Tukia, On two-dimensional quasiconformal groups, Ann. Acad. Sci. Fenn. Ser. A I Math. 5 (1980), 73–78.

    Article  MathSciNet  MATH  Google Scholar 

  78. ___, A quasiconformal group not isomorphic to a Möbius group, Ann. Acad. Sci. Fenn. Ser. A I Math. 6 (1981), 149–160.

    Article  MathSciNet  MATH  Google Scholar 

  79. ___, Quasiconformal extension of quasisymmetric mappings compatible with a Möbius group, Acta Math. 154 (1985), 153–193.

    Article  MathSciNet  MATH  Google Scholar 

  80. P. Tukia and J. Väisälä, Quasiconformal extension from dimension n to n + 1, Ann. of Math. 115 (1982), 331–348.

    Article  MathSciNet  MATH  Google Scholar 

  81. ___, Bilipschitz extensions of maps having quasiconformal extensions, Math. Ann. 269 (1984), 561–572.

    Article  MathSciNet  MATH  Google Scholar 

  82. J. Väisälä, On quasiconformal mappings in space, Ann. Acad. Sci. Fenn. Ser. A I Math. 298 (1961), 1–36.

    MathSciNet  MATH  Google Scholar 

  83. ___, On quasiconformal mappings of a ball, Ann. Acad. Sci. Fenn. Ser. A I Math. 304 (1961), 1–7.

    MathSciNet  MATH  Google Scholar 

  84. ___, Lectures on n-dimensional quasiconformal mappings, Lectures Notes in Math., Vol. 229, Springer-Verlag, 1971.

    Google Scholar 

  85. ___, A survey of quasiregular maps in R n, Proc. Internat. Congr. Math. (Helsinki, 1978), Acad. Sci. Fennica, Helsinki, 1980, pp. 685–691.

    Google Scholar 

  86. ___, Quasimöbius maps, J. Analyse Math. 44 (1984/85), 218–234.

    Article  MathSciNet  MATH  Google Scholar 

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  1. Mathematical Sciences Research Institute, 94720, Berkkeley, California, USA

    F. W. Gehring

  2. University of Michigan, 48109, Ann Arbor, Michigan, USA

    F. W. Gehring

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  1. F. W. Gehring
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Matti Vuorinen

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Gehring, F.W. (1992). Topics in quasiconformal mappings. In: Vuorinen, M. (eds) Quasiconformal Space Mappings. Lecture Notes in Mathematics, vol 1508. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0094236

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