Israel Journal of Mathematics

, Volume 130, Issue 1, pp 323–346 | Cite as

Curvature and uniformization

Article

Abstract

We approach the problem of uniformization of general Riemann surfaces through consideration of the curvature equation, and in particular the problem of constructing Poincaré metrics (i.e., complete metrics of constant negative curvature) by solving the equation Δu-e 2u=Ko(z) on general open surfaces. A few other topics are discussed, including boundary behavior of the conformal factore 2u giving the Poincaré metric when the Riemann surface has smoothly bounded compact closure, and also a curvature equation proof of Koebe's disk theorem.

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References

  1. [1]
    L. Ahlfors,Complex Analysis, McGraw-Hill, New York, 1996.Google Scholar
  2. [2]
    L. Ahlfors,Conformal Invariants, McGraw-Hill, New York, 1973.MATHGoogle Scholar
  3. [3]
    T. Aubin,Nonlinear Analysis on Manifolds. Monge-Ampere Equations, Springer-Verlag, New York, 1982.MATHGoogle Scholar
  4. [4]
    A. Beardon and C. Pommerenke,The Poincaré metric of plane domains, Journal of the London Mathematical Society18 (1979), 475–483.CrossRefMathSciNetGoogle Scholar
  5. [5]
    M. Berger,On Riemannian structures of prescribed Gaussian curvature for compact 2-manifolds, Journal of Differential Geometry5 (1971), 328–332.Google Scholar
  6. [6]
    P. Chrusciel,Semi-global existence and convergence of solutions of the Robinson-Trautman (2-dimensional Calabi) equation, Communications in Mathematical Physics137 (1991), 289–313.MATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    H. Farkas and I. Kra,Riemann Surfaces, second edition, Springer-Verlag, New York, 1992.MATHGoogle Scholar
  8. [8]
    O. Forster,Lectures on Riemann Surfaces, Springer-Verlag, New York, 1981.MATHGoogle Scholar
  9. [9]
    H. Grauert and H. Reckziegel,Hermitesch Metriken und normale Familien holomorpher Abbildunen, Mathematische Zeitschrift89 (1965), 108–125.MATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    R. Hamilton,The Ricci flow on surfaces, Contemporary Mathematics71 (1988).Google Scholar
  11. [11]
    M. Heinz,The conformal mapping of simply connected Riemann surfaces, Annals of Mathematics50 (1949), 686–690.CrossRefMathSciNetGoogle Scholar
  12. [12]
    J. Kazdan and F. Warner,Curvature functions for compact 2-manifolds, Annals of Mathematics99 (1974), 14–47.CrossRefMathSciNetGoogle Scholar
  13. [13]
    S. Kobayashi,Hyperbolic Manifolds and Holomorphic Mappings, M. Dekker, New York, 1970.MATHGoogle Scholar
  14. [14]
    S. Krantz,Complex Analysis: The Geometric Viewpoint, Carus Mathematical Monographs, Series #23, Mathematical Association of America, 1990.Google Scholar
  15. [15]
    R. Mazzeo,Elliptic theory of differential edge operators I, Communications in Partial Differential Equations16 (1991), 1615–1664.MATHMathSciNetCrossRefGoogle Scholar
  16. [16]
    R. Mazzeo,Regularity for the singular Yamabe problem, Indiana University Mathematics Journal40 (1991), 1277–1299.MATHCrossRefMathSciNetGoogle Scholar
  17. [17]
    B. Osgood, R. Phillips and P. Sarnak,Extremals of determinants of Laplacians, Journal of Functional Analysis80 (1988), 212–234.MATHCrossRefMathSciNetGoogle Scholar
  18. [18]
    R. Osserman,Koebe's general uniformization theorem: the parabolic case, Annales Academiae Scientiarum Fennicae. Series A I258 (1958), 1–7.MathSciNetGoogle Scholar
  19. [19]
    E. Picard,De l'equation Δu=ke u sur une surface de Riemann fermée, Journal de Mathématiques (4)9 (1893), 273–291.Google Scholar
  20. [20]
    G. Springer,Introduction to Riemann Surfaces, Addison-Wesley, Reading, Mass., 1957.MATHGoogle Scholar
  21. [21]
    M. Taylor,Partial Differential Equations, Vols. 1–3, Springer-Verlag, New York, 1996.Google Scholar
  22. [22]
    M. Tsuji,Protential theory and Modern Function Theory, Chelsea, New York, 1959.Google Scholar

Copyright information

© Hebrew University 2002

Authors and Affiliations

  1. 1.Department of MathematicsStanford UniversityStanfordUSA
  2. 2.Department of MathematicsUniversity of North CarolinaChapel HillUSA

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