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On the Bergman representative coordinates

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Abstract

We study the set where the so-called Bergman representative coordinates (or Bergman functions) form an immersion. We provide an estimate of the size of a maximal geodesic ball with respect to the Bergman metric contained in this set. By concrete examples we show that these estimates are the best possible.

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References

  1. Berceanu S. On the geometry of complex Grassmann manifold, its noncompact dual and coherent states. Bull Belg Math Soc Simon Stevin, 1997, 4: 205–243

    MATH  MathSciNet  Google Scholar 

  2. Bergman S. Über die Existenz von Repräsentantenbereichen in der Theorie der Abbildung durch Paare von Funktionen zweier komplexen Veränderlichen. Math Ann, 1930, 102: 430–446

    Article  MathSciNet  Google Scholar 

  3. Błocki Z. The Bergman metric and the pluricomplex Green function. Trans Amer Math Soc, 2005, 357: 2613–2625

    Article  MATH  MathSciNet  Google Scholar 

  4. Boas H P. The Lu Qi-Keng conjecture fails generically. Proc Amer Math Soc, 1996, 124: 2021–2027

    Article  MATH  MathSciNet  Google Scholar 

  5. Davidov J. The representative domain of a complex manifold and the Lu-Qi-Keng conjecture. C R Acad Bulgare Sci, 1977, 30: 13–16

    MATH  MathSciNet  Google Scholar 

  6. Dinew Ż. An example for the holomorphic sectional curvature of the Bergman metric. Ann Polon Math, 2010, 98: 147–167

    Article  MATH  MathSciNet  Google Scholar 

  7. Herbort G. On the geodesics of the Bergman metric. Math Ann, 1983, 264: 39–51

    Article  MATH  MathSciNet  Google Scholar 

  8. Kató S. Canonical domains in several complex variables. Pacific J Math, 1967, 21: 279–291

    MATH  MathSciNet  Google Scholar 

  9. Kim K T, Krantz S G. The Bergman metric invariants and their boundary behavior. Explorations in complex and Riemannian geometry ser. Contemp Math, 332. Providence RI: Amer Math Soc, 2003, 139–151

    Google Scholar 

  10. Kobayashi S. Geometry of bounded domains. Trans Amer Math Soc, 1959, 92: 267–290

    Article  MATH  MathSciNet  Google Scholar 

  11. Krantz S G. Geometric Analysis and Function Spaces. CBMS Regional Conference Series in Mathematics. Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1993

    Google Scholar 

  12. Krantz S G, Yu J. On the Bergman invariant and curvatures of the Bergman metric. Illinois J Math, 1996, 40: 226–244

    MATH  MathSciNet  Google Scholar 

  13. Leichtweiss K. Zur Riemannschen Geometrie in Grassmannschen Mannigfaltigkeiten. Math Z, 1961, 76: 334–366

    Article  MATH  MathSciNet  Google Scholar 

  14. Lewittes J. Differentials and metrics on Riemann surfaces. Trans Amer Math Soc, 1969, 139: 311–318

    MATH  MathSciNet  Google Scholar 

  15. Lu Q K. On Kaehler manifolds with constant curvature. Acta Math Sin, 1966, 16: 269–281

    MATH  Google Scholar 

  16. Lu Q K. Holomorphic invariant forms of a bounded domain. Sci China Ser A, 2008, 51: 1945–1964

    Article  MATH  MathSciNet  Google Scholar 

  17. Lu Q K. The conjugate points of ℂℙ∞ and the zeroes of Bergman kernel. Acta Math Sci Ser B Engl Ed, 2009, 29: 480–492

    MATH  MathSciNet  Google Scholar 

  18. Rosenthal P. On the zeros of the Bergman function in doubly-connected domains. Proc Amer Math Soc, 1969, 21: 33–35

    Article  MATH  MathSciNet  Google Scholar 

  19. Skwarczyński M. The invariant distance in the theory of pseudoconformal transformations and the Lu Qi-keng conjecture. Proc Amer Math Soc, 1969, 22: 305–310

    MATH  MathSciNet  Google Scholar 

  20. Xu Y. Theory of Complex Homogeneous Bounded Domains. ser. Mathematics and its Applications. Beijing: Science Press, 2005

    Google Scholar 

  21. Zwonek W. Asymptotic behavior of the sectional curvature of the Bergman metric for annuli. Ann Pol Math, 2010, 98: 291–299

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Jagiellonian University, Kraków, 30-348, Poland

    Żywomir Dinew

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  1. Żywomir Dinew
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Dinew, Ż. On the Bergman representative coordinates. Sci. China Math. 54, 1357–1374 (2011). https://doi.org/10.1007/s11425-011-4243-4

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  • DOI: https://doi.org/10.1007/s11425-011-4243-4

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