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A canonical metric for Möbius structures and its applications

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References

  • [A] Apanasov, B.N.: Deformations of Conformal Structures on Hyperbolic Manifolds. Math. Sci. Res. Inst. (Preprint 1990)

  • [B] Beardon, A.F.: Geometry of Discrete Groups. (Grad. Texts Math., vol. 91) Berlin Heidelberg New York: Springer 1982

    Google Scholar 

  • [Bl] Blaschke, W.: Vorlesungen über Differential-Geometrie, vol. 3. Berlin Heidelberg New York: Springer 1929

    Google Scholar 

  • [EM] Epstein, D.B.A., Marden, A.: Convex Hulls in Hyperbolic Spaces. A theorem of Sullivan, and Measured Pleated Surfaces. In: Epstein, D.B.A. (ed.) Symposium on Hyperbolic Geometry, Kleinian Groups, and 3-dimensional Topology. (Lond. Math. Soc. Lect. Notes., vol. 111, pp. 113–253) Cambridge: Cambridge University Press 1987

    Google Scholar 

  • [F1] Lelong-Ferrand, J.: Transformations conformes et quasi-conformes des varietes riemanniennes. Acad. R. Belg. Sci. Mem. Coll.8(2), 39 (1971)

    Google Scholar 

  • [F2] Ferrand, J.: A characterization of quasiconformal mappings by the behavior of a function of three points. In: Laine, I., Rickman, S., Sorvali, T. (eds.), Complex Analysis, Joensu 1987. (Lect. Notes Math., vol. 1351, pp. 110–123) Berlin Heidelberg New York: Springer 1988

    Chapter  Google Scholar 

  • [GP] Gehring, F., Palka, B.: Quasiconformally homogeneous domains. J. Anal. Math.30, 172–199 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  • [G] Goldman, W.M.: Projective Structures with Fuchsian Holonomy. J. Differ. Geom.25, 297–326 (1987)

    MATH  Google Scholar 

  • [Gu] Gunning, R.C.: Special coordinate coverings of Riemann surfaces. Math. Ann.170, 67–86 (1967)

    Article  MATH  MathSciNet  Google Scholar 

  • [Ka] Kamishima, Y.: Conformally flat manifolds whose developments are not surjective. I. Trans. Am. Math. Soc.294, 607–623 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  • [K] Klingenberg, W.: Riemannian Geometry. (De Gruyter Stud. Math., vol. 1) Berlin New York: de Gruyter 1982

    MATH  Google Scholar 

  • [Kob] Kobayashi, S.: Hyperbolic Manifolds and Holomorphic Mappings. (Pure Appl. Math. Monogr., vol. 2) New York: Dekker 1970

    MATH  Google Scholar 

  • [Kup] Kuiper, N.H.: On conformally-flat spaces in the large. Ann. Math.50, 916–924 (1949)

    Article  MathSciNet  Google Scholar 

  • [Kul1] Kulkarni, R.S.: On the Principle of Uniformization. J. Differ. Geom.13, 109–138 (1978)

    MATH  MathSciNet  Google Scholar 

  • [Kul2] Kulkarni, R.S.: Conformal Structures and Möbius Structures. In: Kulkarni, R.S., Pinkall, U. (eds.) Conformal Geometry. (Aspects Math., vol. 12, pp. 1–39) Braunschweig Wiesbaden: Vieweg 1988

    Google Scholar 

  • [KP] Kulkarni, R.S., Pinkall, U.: Uniformizations of geometric structures with applications to conformal geometry. In: Naveira, A.M. et al. (eds.) Differential Geometry Peñiscola 1985. (Lect. Notes Math., vol. 1209, pp. 190–209) Berlin Heidelberg New York: Springer 1986

    Chapter  Google Scholar 

  • [Kr1,2] Kra, I.: Deformations of fuchsian groups, I and II. Duke Math. J.36, 537–546 (1969); ibid.38, 499–508 (1971)

    Article  MATH  MathSciNet  Google Scholar 

  • [L1] Lafontaine, J.: Module de structures conformes plates et cohomologie de groupes discreres. C.R. Acad. Sci., Paris297, 655–658 (1983)

    MATH  MathSciNet  Google Scholar 

  • [L2] Lafontaine, J.: The theorem of Lelong-Ferrand and Obata. In: Kulkarni, R.S., Pinkall, U. (eds.) Conformal Geometry. (Aspects Math., vol. 12, pp. 93–103) Braunschweig Wiesbaden: Vieweg 1988

    Google Scholar 

  • [Le] Leutwiler, H.: A Riemannian metric invariant under Möbius transformations inR n. In: Laine, I., Rickman, S., Sorvali, T. (eds.) Complex Analysis, Joensu 1987. (Lect. Notes Math., vol. 1351, pp. 223–235) Berlin Heidelberg New York: Springer 1988

    Chapter  Google Scholar 

  • [LN] Loewner, C., Nirenberg, L.: Partial Differential Equations Invariant under Conformal or Projective Transformations. In: Ahlfors, L.V., Kra, I., Maskit, B., Nirenberg, L. (eds.) Contributions to Analysis, pp. 245–272. New York London: Academic Press 1974

    Google Scholar 

  • [M] Martin, G.: Quasiconformal and bi-Lipschitz homeomorphisms, uniform domains and the quasihyperbolic metric. Trans. Am. Math. Soc.292, 169–191 (1985)

    Article  MATH  Google Scholar 

  • [Mas] Maskit, B.: On a Class of Kleinian Groups. Ann. Acad. Sci. Fenn. Ser. A442, 1–8 (1969)

    MathSciNet  Google Scholar 

  • [Mi] Millson, J.: A remark on Raghunathan's vanishing theorem. Topology24, 495–498 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  • [MS] Morgan, J., Shalen, P.: Valuations, Trees, and Degenerations of Hyperbolic Structures. I. Ann. Math.120, 401–476 (1984)

    Article  MathSciNet  Google Scholar 

  • [O1,2] Obata, M.: I. Conformal transformations of Riemannian manifolds. J. Differ. Geom.4, 311–333 (1970); 2. The conjectures about conformal transformations. ibid.6, 247–258 (1971)

    MATH  MathSciNet  Google Scholar 

  • [On] O'Neill, B.: Semi-Riemannian Geometry with applications to relativity. New York London: Academic Press 1983

    MATH  Google Scholar 

  • [P] Penner, R.: An Introduction to Train-tracks. In: Epstein, D.B.A. (ed.), Symposium on Hyperbolic Geometry Kleinian Groups, and 3-dimensional Topology. (Lond. Math. Soc. Lect. Notes, vol. 112, pp. 77–90) Cambridge: Cambridge University Press 1986

    Google Scholar 

  • [Pom] Pommerenke, C.: Univalent Functions. Göttingen: Vadenhoeck and Ruprecht 1975

    MATH  Google Scholar 

  • [S] Spivak, M.: A Comprehensive Introduction to Differential Geometry, vol. 3, Berkeley: Publish or Perish 1975

    MATH  Google Scholar 

  • [T] Thurston, W.: The Geometry and Topology of 3-Manifolds. Princeton: Princeton University Mathematics Department 1979

    Google Scholar 

  • [W] Whitney, H.: Geometric Integration Theory. Princeton: Princeton University Press 1966

    Google Scholar 

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Partially supported by an NSF grant, and a PSC-CUNY award

Partially supported by a DFG-Project Pi 158/2-1

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Kulkarni, R.S., Pinkall, U. A canonical metric for Möbius structures and its applications. Math Z 216, 89–129 (1994). https://doi.org/10.1007/BF02572311

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