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Quasiconformal mappings and manifolds of negative curvature

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Curvature and Topology of Riemannian Manifolds

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References

  1. L. AHLFORS, Zum theorie der überlagerungsflächen, Acta Math. 5, 157–194 (1936).

    MathSciNet  MATH  Google Scholar 

  2. L. AHLFORS, On quasiconformal mappings, J. d'Analyse Math. 3, 1–58 (1954).

    Article  MathSciNet  MATH  Google Scholar 

  3. L. AHLFORS, On quasiconformal mappings, J. d'Analyse Math. 4, 204–208 (1954).

    MathSciNet  MATH  Google Scholar 

  4. P. BERARD-D. MEYER, Inégalités isopérimétriques et applications, Ann. Sci. de l'E. N. S. de Paris 13, 513–542 (1982).

    MathSciNet  MATH  Google Scholar 

  5. L. BERS, Quasiconformal mappings and Teichmüller's theorem, 89–119 in "Analytic functions", Princeton Math. Series Vol 24, Princeton Univ. Press, Princeton (1960).

    Google Scholar 

  6. D. BURNS Jr-S. SHNIDER, Geometry of hypersurfaces and mapping theorems in C n, Comment. Math. Helvetici 54, 199–217 (1979).

    Article  MathSciNet  MATH  Google Scholar 

  7. P. EBERLEIN-B. O'NEILL, Visibility manifolds, Pacific J. Math. 46, 45–109 (1973).

    Article  MathSciNet  MATH  Google Scholar 

  8. I. S. GAL, Conformally invariant metrics on Riemann surfaces, Proc. Nat.Acad. Sci.USA 45, 1629–1633 (1959).

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  10. F.W. GEHRING, Extremal length definitions for the conformal capacity of rings in space, Michigan Math. J. 9, 137–150 (1962).

    Article  MathSciNet  MATH  Google Scholar 

  11. F.W. GEHRING, Rings and Quasiconformal mappings in space, Trans. Amer. Math. Soc. 103, 353–393 (1962).

    Article  MathSciNet  MATH  Google Scholar 

  12. F.W. GEHRING-B.G. OSGOOD, Uniform domains and the quasihyperbolic metric, J. d'Analyse Math. 36, 50–74 (1979).

    Article  MathSciNet  MATH  Google Scholar 

  13. F.W. GEHRING-B.P. PALKA, Quasiconformally homogeneous domains, J. d'Analyse Math. 30, 172–199 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  14. F.W. GEHRING-J. VÄISÄLÄ, On the geometric definition for quasiconformal mappings, Comment. Math. Helvetici 36, 19–32 (1961).

    Article  MATH  Google Scholar 

  15. M. GROMOV, Hyperbolic manifolds, groups and actions, 183–213 in "Riemann surfaces and related topics, Stony Brook 1978", Ann. of Math. Studies Vol. 97, Princeton University Press, Princeton (1981).

    Google Scholar 

  16. M. GROMOV, Structures métriques pour les variétés riemanniennes, chap. 6, notes de cours rédigées par J. Lafontaine et P. Pansu, CEDIC-Fernand-Nathan, Paris (1981).

    Google Scholar 

  17. M. GROMOV, Filling Riemannian manifolds, J. Diff. Geo. 18, 1–148 (1983).

    MathSciNet  MATH  Google Scholar 

  18. M. GROMOV, Pseudo-holomorphic curves in symplectic manifolds, to appear in Inventiones Mathematicae.

    Google Scholar 

  19. H. GRÖTZSCH, Über die Verzerrung bei schlichten nichtkonformen Abbildungen und über eine damit zusammenhängende Erweiterung des Picardschen Satzes, Ber. Verh. Sächs. Akad. Wiss. Leipzig 80, 503–507 (1928).

    MATH  Google Scholar 

  20. U. HAAGERUP-H.J. MUNKHOLM, Simplices of maximal volume in hyperbolic n-space, Acta Math. 147, 1–11 (1981).

    Article  MathSciNet  MATH  Google Scholar 

  21. E. HEINTZE On homogeneous manifolds of negative curvature, Math. Ann. 211, 23–24 (1974).

    Article  MathSciNet  MATH  Google Scholar 

  22. J.A. JENKINS, A new criterion for quasiconformal mappings, Ann. Of Math. 65, 208–214 (1957).

    Article  MathSciNet  MATH  Google Scholar 

  23. D. JERISON-J.M. LEE, The Sobolev inequality on the Heisenberg group and the Yamabe problem on CR manifolds, Séminaire Goulaouic-Meyer-Schwarz, Ecole Polytechnique, Palaiseau (1984).

    MATH  Google Scholar 

  24. M. KANAI, Rough isometries and the parabolicity of Riemannian manifolds, Preprint Keio Univ. Yokohama (1985).

    Google Scholar 

  25. S. KOBAYASHI, Hyperbolic manifolds and holomorphic mappings, Dekker, New-York (1970).

    MATH  Google Scholar 

  26. S. KOBAYASHI, Invariant distances for projective structures, Symp. Math. 26, 153–161 (1982).

    MathSciNet  MATH  Google Scholar 

  27. A. KORANYI-H.M. REIMANN, Quasiconformal mappings on the Heisenberg group, Preprint Bern (1984)

    Google Scholar 

  28. N.H. KUIPER, Einstein spaces and connections I, Proc. Kon. Akad. v. Wetensch. Amsterdam 12, 506–521 (1950).

    MATH  Google Scholar 

  29. M.A. LAVRENTIEV, Sur une classe de représentation continue, Math. Sb. 42, 407–423 (1935).

    Google Scholar 

  30. O.LEHTO-K.I. VIRTANEN, Quasikonforme Abbildungen, Grundlehren der Math... Band 126, Springer Verlag Berlin...(1965).

    Google Scholar 

  31. J. LELONG-FERRAND, Invariants conformes globaux sur les variétés riemanniennes, J. of Diff. Geo. 8, 487–510 (1973).

    MathSciNet  MATH  Google Scholar 

  32. J. LELONG-FERRAND, Etude d'une classe d'applications liées à des homomorphismes d'algèbres de fonctions, et généralisant les quasiconformes, Duke Math. J. 40, 163–185 (1973)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  34. Ch. LOEWNER, On the conformal capacity in space, J. Math. Mech. 8, 411–414 (1959).

    MathSciNet  MATH  Google Scholar 

  35. Ch. LOEWNER-L. NIRENBERG, Partial Differential Equations invariant under Conformal or Projective Transformations, 245–272 in "Contributions to Analysis", ed. L. Ahlfors, I. Kra, B. Maskit, L. Nirenberg, Academic Press, New-York (1974).

    Google Scholar 

  36. O. MARTIO-S. RICKMAN-J. VÄISÄLÄ, Definitions for quasiregular mappings, Ann. Acad. Sci. Fennicae Ser. A.I 488, 1–40 (1969).

    MATH  Google Scholar 

  37. C.B. MORREY, On the solution of quasilinear elliptic partial differential equations, Trans. Amer. Math. Soc. 43, 126–166 (1938).

    Article  MathSciNet  MATH  Google Scholar 

  38. M. MORSE, Recurrent geodesics on a surface of negative curvature, Trans. Amer. Math. Soc. 2 84–100 (1921).

    Article  MathSciNet  MATH  Google Scholar 

  39. G.D. MOSTOW, Quasiconformal mappings in n-space and the rigidity of hyperbolic space forms, Publ. I.H.E.S. 34, 53–104 (1968).

    Article  MathSciNet  MATH  Google Scholar 

  40. G.D. MOSTOW, Strong rigidity of discrete subgroups and quasiconformal mappings over a division algebra, 203–209 in proc. of "Discrete subgroups of Lie groups and application to moduli, Bombay 1973".

    Google Scholar 

  41. G.D. MOSTOW, Strong rigidity of locally symmetric spaces, Annals of Math. Studies Vol 78, Princeton Univ. Press, Princeton (1973).

    MATH  Google Scholar 

  42. M. OBATA, The conjectures on conformal transformations of Riemannian manifolds, Bull. Amer. Math. Soc. 77, 269–270 (1971).

    Article  MathSciNet  MATH  Google Scholar 

  43. P. PANSU, An isoperimetric inequality on the Heisenberg group, in proc. of "Differential geometry on homogeneous spaces, Torino (1983).

    Google Scholar 

  44. P. PANSU, Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un, Preprint Ecole Polytechnique, Palaiseau (1984).

    Google Scholar 

  45. I.N. PESIN, Metric properties of Q-quasikonformal mappings, Math. Sb. 40 (82), 281–295 (1956).

    MathSciNet  Google Scholar 

  46. A. PFLUGER, Quasikonforme Abbildungen and logarithmische Kapazität, Ann. Institut Fourier 2, 29–80 (1951).

    MathSciNet  Google Scholar 

  47. G. POLYA-G. SZEGÖ, Isoperimetric inequalities in mathematical physics, Ann. of Math. Studies Nr 27, Princeton Univ. Press (1951).

    Google Scholar 

  48. I.G. RESHETNJAK, On conformal mappings of a space, Dokl. Akad. Nauk. SSSR 130, 1196–1198 (1960) = Soviet. Math. Dokl. 1, 153–155 (1960).

    MathSciNet  Google Scholar 

  49. I.G. RESHETNJAK, Stability of conformal mappings in multi-dimensional spaces, Sibirsk. Mat. Zh. 8, 91–114 (1967).

    MathSciNet  Google Scholar 

  50. S. RICKMAN, On the number of omitted values of entire quasiregular mappings, J. d'Analyse Math. 37, 100–117 (1980).

    Article  MathSciNet  MATH  Google Scholar 

  51. S. RICKMAN, Quasiregular mappings and metrics on the n-sphere with punctures, Comment. Math. Helvetici 59, 136–148 (1984).

    Article  MathSciNet  MATH  Google Scholar 

  52. S. RICKMAN, The analogue of Picard's theorem for quasiregular mappings in dimension three, to appear in Acta Mathematica.

    Google Scholar 

  53. D. SULLIVAN, On the ergodic theory at infinity of an arbitrary discrete group of hyperbolic motions, 465–496 in "Riemann surfaces and related topics, Stony Brook 1978", Ann. of Math. Studies Vol.97, Princeton Univ. Press, Princeton (1981).

    Google Scholar 

  54. H. TANAKA, Boundary behaviours of quasiregular mappings, 405–408 in "Complex Analysis, Joensuu 1978", Lecture Notes Vol. 747, Springer Verlag Berlin... (1979).

    Google Scholar 

  55. O. TEICHMÜLLER, Extremale quasikonforme Abbildungen und quadratische Differentialen, Abh. Preuss. Akad. Wiss. 22, 1–197 (1940).

    Google Scholar 

  56. W. THURSTON, Geometry and topology of 3-manifolds, Preprint Princeton (1978).

    Google Scholar 

  57. P. TUKIA, On two-dimensional quasiconformal groups, Ann. Acad. Sci. Fennicae Ser. A I 5, 73–78 (1980)

    MathSciNet  MATH  Google Scholar 

  58. P. TUKIA, A quasiconformal group not isomorphic to a Möbius group, Ann. Acad. Sci. Fennicae Ser. A.I 6, 149–160 (1981).

    MathSciNet  MATH  Google Scholar 

  59. P. TUKIA, On quasiconformal groups, Preprint Helsinki (1982).

    Google Scholar 

  60. P. TUKIA, Differentiability and rigidity of Möbius groups, preprint Univ. Helsinki (1982).

    Google Scholar 

  61. P. TUKIA, Quasiconformal extension of quasisymmetric mappings compatible with a Möbius group, Acta Math. 154, (1985).

    Google Scholar 

  62. P. TUKIA-J. VÄISÄLÄ, A remark on 1-quasiconformal maps, Ann. Acad. Sci. Fennicae ser. A I 10, 839–842 (1985).

    MathSciNet  MATH  Google Scholar 

  63. J. VÄISÄLÄ, Lectures on n-dimensional quasiconformal mappings, Lecture Notes in Math. Vol. 229, Springer Verlag, Berlin... (1970)/

    MATH  Google Scholar 

  64. N. VAROPOULOS, Chaînes de Markov et inégalités isopérimétriques, C. R. Acad. Sci. Paris 298 (10), 233–236 (1984).

    MathSciNet  MATH  Google Scholar 

  65. N. VAROPOULOS, Théorie du potentiel sur les groupes nilpotents, C. R. Acad. Sci. Paris 301 (5), 143–144 (1985).

    MathSciNet  MATH  Google Scholar 

  66. E. VESENTINI, Invariant metrics on convex cones, Ann. Scu. Norm. Sup. Pisa Ser. 4 3, 671–696 (1976).

    MathSciNet  MATH  Google Scholar 

  67. M. VILLE, Sur le volume des variétés riemanniennes pincées, to appear in Bull. Soc. Math. de France (1985).

    Google Scholar 

  68. M. VUORINEN, Growth inequalities for quasiregular mappings, preprint Univ. Helsinki (1979).

    Google Scholar 

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  1. Centre de Mathématiques Ecole Polytechnique, U.A. 169 du CNRS, F-91128, Palaiseau Cédex

    Pierre Pansu

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  1. Pierre Pansu
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Katsuhiro Shiohama Takashi Sakai Toshikazu Sunada

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Pansu, P. (1986). Quasiconformal mappings and manifolds of negative curvature. In: Shiohama, K., Sakai, T., Sunada, T. (eds) Curvature and Topology of Riemannian Manifolds. Lecture Notes in Mathematics, vol 1201. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0075658

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  • DOI: https://doi.org/10.1007/BFb0075658

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