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Curvature and uniformization

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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. L. Ahlfors,Complex Analysis, McGraw-Hill, New York, 1996.

    Google Scholar 

  2. L. Ahlfors,Conformal Invariants, McGraw-Hill, New York, 1973.

    MATH  Google Scholar 

  3. T. Aubin,Nonlinear Analysis on Manifolds. Monge-Ampere Equations, Springer-Verlag, New York, 1982.

    MATH  Google Scholar 

  4. A. Beardon and C. Pommerenke,The Poincaré metric of plane domains, Journal of the London Mathematical Society18 (1979), 475–483.

    Article  MathSciNet  Google Scholar 

  5. M. Berger,On Riemannian structures of prescribed Gaussian curvature for compact 2-manifolds, Journal of Differential Geometry5 (1971), 328–332.

    Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  7. H. Farkas and I. Kra,Riemann Surfaces, second edition, Springer-Verlag, New York, 1992.

    MATH  Google Scholar 

  8. O. Forster,Lectures on Riemann Surfaces, Springer-Verlag, New York, 1981.

    MATH  Google Scholar 

  9. H. Grauert and H. Reckziegel,Hermitesch Metriken und normale Familien holomorpher Abbildunen, Mathematische Zeitschrift89 (1965), 108–125.

    Article  MATH  MathSciNet  Google Scholar 

  10. R. Hamilton,The Ricci flow on surfaces, Contemporary Mathematics71 (1988).

  11. M. Heinz,The conformal mapping of simply connected Riemann surfaces, Annals of Mathematics50 (1949), 686–690.

    Article  MathSciNet  Google Scholar 

  12. J. Kazdan and F. Warner,Curvature functions for compact 2-manifolds, Annals of Mathematics99 (1974), 14–47.

    Article  MathSciNet  Google Scholar 

  13. S. Kobayashi,Hyperbolic Manifolds and Holomorphic Mappings, M. Dekker, New York, 1970.

    MATH  Google Scholar 

  14. S. Krantz,Complex Analysis: The Geometric Viewpoint, Carus Mathematical Monographs, Series #23, Mathematical Association of America, 1990.

  15. R. Mazzeo,Elliptic theory of differential edge operators I, Communications in Partial Differential Equations16 (1991), 1615–1664.

    Article  MATH  MathSciNet  Google Scholar 

  16. R. Mazzeo,Regularity for the singular Yamabe problem, Indiana University Mathematics Journal40 (1991), 1277–1299.

    Article  MATH  MathSciNet  Google Scholar 

  17. B. Osgood, R. Phillips and P. Sarnak,Extremals of determinants of Laplacians, Journal of Functional Analysis80 (1988), 212–234.

    Article  MATH  MathSciNet  Google Scholar 

  18. R. Osserman,Koebe's general uniformization theorem: the parabolic case, Annales Academiae Scientiarum Fennicae. Series A I258 (1958), 1–7.

    MathSciNet  Google Scholar 

  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. G. Springer,Introduction to Riemann Surfaces, Addison-Wesley, Reading, Mass., 1957.

    MATH  Google Scholar 

  21. M. Taylor,Partial Differential Equations, Vols. 1–3, Springer-Verlag, New York, 1996.

    Google Scholar 

  22. M. Tsuji,Protential theory and Modern Function Theory, Chelsea, New York, 1959.

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Stanford University, 94305, Stanford, CA, USA

    Rafe Mazzeo

  2. Department of Mathematics, University of North Carolina, 27514, Chapel Hill, NC, USA

    Michael Taylor

Authors
  1. Rafe Mazzeo
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  2. Michael Taylor
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Corresponding author

Correspondence to Rafe Mazzeo.

Additional information

Research supported in part by NSF Grant DMS-9971975 and also at MSRI by NSF grant DMS-9701755.

Research supported in part by NSF Grant DMS-9877077

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Mazzeo, R., Taylor, M. Curvature and uniformization. Isr. J. Math. 130, 323–346 (2002). https://doi.org/10.1007/BF02764082

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  • DOI: https://doi.org/10.1007/BF02764082

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