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On the Uniformization of Parabolic Manifolds

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Contributions to Several Complex Variables

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Abstract

The concept of an open parabolic Riemann surface is classical. It is well-known that there are many different but equivalent characterizations, each of which can be generalized in some way to complex manifolds of higher dimension. However, as is to be expected, these generalized concepts are in general not equivalent. In this article parabolicity is defined via the existence of an (unbounded) exhaustion satisfying the homogeneous complex Monge-Ampere equation which, in the case of Riemann surfaces is simply the Laplace equation. There are several reasons for choosing this as the definition of parabolic manifolds. First of all it includes affine algebraic manifolds and with this definition, the classical value distribution theory over ℂn can be extended to parabolic manifolds in such a way that the defect relation is still intrinsic (cf. section 1 below). Secondly, this definition is indeed quite intrinsic as it is possible to obtain (under appropriate assumptions on the regularity of the exhaustion function) uniformization theorems for parabolic manifolds. Thirdly, by allowing bounded exhaustions (so that the manifolds under consideration are no longer parabolic but rather, Kobayashi hyperbolic), it is discovered that the Monge-Ampere condition is intrinsically related to the condition of certain complex curves being geodesics(or extremal disc) of the Kobayashi metric (cf. sections 4 and 5).

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References

  1. Ahlfors, L., The theory of meromorphic curves, Acta Soc. Fenn. Nova Ser. A 3 (4), (1941).

    Google Scholar 

  2. Alexander, H., Projective Capacity, Recent Development in Several Complex Variables, edited by J. Fornaess, Ann. Math Studies 100 (1981).

    Google Scholar 

  3. Bedford, E., and Kalka, M., Foliations and complex Monge-Ampere equations, Comm. Pure and Appl. Math, 30, 543–571 (1977).

    Article  MathSciNet  MATH  Google Scholar 

  4. Bedford, E. and Taylor, B.A., Variational properties of the complex Monge-Ampere equation II, intrinsic norms, Am. J. Math 101, 1131–1166 (1977).

    Article  MathSciNet  Google Scholar 

  5. Bland, J., Duchamp, T. and Kalka, M., On the automorphism group of strictly convex domains in Tn, preprint, (1984).

    Google Scholar 

  6. Burns, D., Curvature of Monge-Ampere foliations and parabolic•manifolds, Ann. of Math. 115, 349–373 (1982).

    Article  MathSciNet  MATH  Google Scholar 

  7. Carlson, J., and Griffiths, P., A defect relation for equidimensional holomorphic mappings between algebraic varieties, Ann. of Math. 95, 557–584. (1972).

    Article  MathSciNet  MATH  Google Scholar 

  8. Chern, S.S., Complex manifolds without potential theory.

    Google Scholar 

  9. Chern, S.S., Holomorphic curves in the plane, Diff. Geometry in honor of K. Yano, Kinokuniya, Tokyo, 73–94 (1972).

    Google Scholar 

  10. Foote, R., Curvature estimates for Monge-Ampère foliations, Univ. of Michigan Thesis (1983).

    Google Scholar 

  11. Greene, R.E., and Wu, H., Function theory on manifolds which possess a pole, Springer-Verlaa Lecture note series, Vol. 699(1979).

    Google Scholar 

  12. Griffiths, P. and King, J., Nevanlinna theory and holomorphic mappings between algebraic varieties, Acta Math. 130, 145–220 (1973).

    Article  MathSciNet  MATH  Google Scholar 

  13. Harvey, R. and Wells, R.O., Zero sets of non-negative strictly plurisubharmonic functions, Math. Ann. 201, 165–170 (1973).

    Article  MathSciNet  MATH  Google Scholar 

  14. Lempert, L., La métrique de Kobayashi et la representation des domains sur la boule, Bull. Soc. Math, France 109, 427–474 (1981).

    MathSciNet  MATH  Google Scholar 

  15. Lempert, L., Intrinsic norms, Proc. Symp, Pure Math., Vol. 41, 147–150 (1984)

    Article  MathSciNet  Google Scholar 

  16. Leung, K.L., Patrizio, G. and Wong, P.M., Isometries of intrinsic metrics on strictly convex domains, preprint (1985).

    Google Scholar 

  17. Noguchi, J., Holomorphic curves in algebraic varieties, Hiroshima Math. J. 7, 833–853 (1977); Supplement ibid 10, 229–231 (1980).

    MathSciNet  Google Scholar 

  18. Patrizio, G., Parabolic exhaustions for strictly convex domains, Manuscripts Math. 47 271–309 (1984).

    Article  MathSciNet  MATH  Google Scholar 

  19. Patrizio, G., A characterization of complex manifolds biholomorphic to a circular domain, to appear in Math Z.

    Google Scholar 

  20. Patrizio, G. and Wong, P.M., Stability of the Monge-Ampere foliation, Math. Ann. 263, 13–29 (1983).

    Article  MathSciNet  MATH  Google Scholar 

  21. Royden, H. and Wong, P.M., Caratheodory and Kobayashi metrics on convex domains, preprint, (1984).

    Google Scholar 

  22. Shiffman, B., On holomorphic curves and meromorphic maps in projective space, Indiana Univ. Math. J. 28, 627–641 (1979).

    MathSciNet  MATH  Google Scholar 

  23. Sibony, N. and Wong, P.M., Some results on global analytic sets. Sem. P. Lelong/H. Skoda, années 1978–79, Springer Lecture Notes Vol. 822, 221–237.

    Google Scholar 

  24. Sibony, N. and Wong, P.M., Some remarks on the Casorati Weierstrass theorem, Ann. Polon. Math XX XIX, (1991).

    Google Scholar 

  25. Siu, Y.T. and Yau, S.T., Complete Kähler manifolds with non-positive curvature of faster than quadratic decay. Ann. of Math. 105, 225–264 (1977).

    Article  MathSciNet  MATH  Google Scholar 

  26. Stoll, W., Die beiden Hauptsätze der Wertverteilungstheorie bei Funktionen mehrerer komplexer Veränderlichen I, Acta Math. 90, 1–115 (1950) II ibid 92, 55–169, 1954.

    Article  MathSciNet  Google Scholar 

  27. Stoll, W., Mehrfache Integrale auf komplexan Mannigfaltigkeiten. Math. Z. 57, 116–154 (1952).

    Article  MathSciNet  MATH  Google Scholar 

  28. Stoll, W., About the value distribution of holomorphic maps into projective space, Acta Math. 123, 83–114, (1969).

    Article  MathSciNet  MATH  Google Scholar 

  29. Stoll, W., Value distribution on parabolic spaces, Springer Lecture Notes Vol. 600 (1977).

    Google Scholar 

  30. Stoll, W., The characterization of strictly parabolic manifolds, Ann. Scuola Norm. Sup. Pisa, VII, 87–154,(1980)

    MathSciNet  Google Scholar 

  31. Stoll, W., The characterization of strictly parabolic spaces, Compositio Math. 44, 305–373 (1981).

    MathSciNet  MATH  Google Scholar 

  32. Stoll, W., The Ahlfors-Weyl theory of meromorphic maps on parabolic manifolds, Springer Lecture Notes 981, 101–219 (1983).

    MathSciNet  Google Scholar 

  33. Vitter, A., The lemma of the logarithmic derivative in several complex variables, Duke Math. J. 44, 89–104, (1977).

    MathSciNet  MATH  Google Scholar 

  34. Weyl, H. and Weyl, J., Meromorphic functions and analytic curves, Princeton University Press, (1943).

    Google Scholar 

  35. Wong, P.M., Defect relations for maps on parabolic spaces and Kobayashi metrics on projective spaces omitting hyperplanes, Notre Dame Thesis (1976).

    Google Scholar 

  36. Wong, P.M., Geometry of the complex homogeneous MonoeAmpere equation, Invent. Math 67, 261–274 (1982).

    Article  MathSciNet  MATH  Google Scholar 

  37. Wong, P.M., On umbilical hypersurfaces and uniformization of circular domains, Proc. Symp., Pure Math, Vol. 41, 225–252 (1984).

    Google Scholar 

  38. Wu, H., The equidistribution theory of holomorphic curves, Ann. of Math. Studies 64, Princeton Univ. Press (1970).

    Google Scholar 

  39. Wu, H., Mappings of Riemann surfaces (Navanlinna theory), Proc. Symp. Pure Math, 11, 480–532 (1968).

    Article  Google Scholar 

  40. Yau, S.T., On the Ricci curvature of compact Kähler manifolds and complex Monge-Ampere equations I, Comm. Pure and Appl. Math., 31, 339–411 (1978).

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Notre Dame, USA

    Pit-Mann Wong

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  1. Pit-Mann Wong
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Editors and Affiliations

  1. Department of Mathematics, University of Notre Dame, Post Office Box 398, Notre Dame, Indiana, 46556, USA

    Alan Howard (Professors) & Pit-Mann Wong (Professors) (Professors) &  (Professors)

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© 1986 Springer Fachmedien Wiesbaden

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Wong, PM. (1986). On the Uniformization of Parabolic Manifolds. In: Howard, A., Wong, PM. (eds) Contributions to Several Complex Variables. Vieweg+Teubner Verlag, Wiesbaden. https://doi.org/10.1007/978-3-663-06816-7_15

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  • DOI: https://doi.org/10.1007/978-3-663-06816-7_15

  • Publisher Name: Vieweg+Teubner Verlag, Wiesbaden

  • Print ISBN: 978-3-528-08964-1

  • Online ISBN: 978-3-663-06816-7

  • eBook Packages: Springer Book Archive

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