On the Representative Domain

  • Qi-keng Lu


Professor Stefan Bergman [1] introduced the idea of Repräsentantenbereich of a bounded domain Open image in new window in ℂn. However the strict definition of a representative domain is not very clear as noted in [2]. It seems that he called the image f( Open image in new window ) of the mapping f:

$$ {\text{z}} \to \frac{{\partial \log \frac{{{\text{k}}({\text{z}},\bar t)}} {{{\text{k}}({\text{t}},\bar t)}}}} {{\partial \bar t^\beta }}{\text{T}}^{\bar \beta \alpha } ({\text{t}},\bar t) $$
the representative domain of Open image in new window . Here we use the summation convention and denote by \( {\text{k}}({\text{z}},\bar t) \) the Bergman kernel of Open image in new window and by \( T^{\overline \beta \alpha } (t,\overline t ) \) the elements of the inverse matrix of the Bergman metric tensor
$$ T_{\alpha \overline \beta } (t,\overline t ) = \frac{{\partial ^2 \log \,K(t,\overline t )}}{{\partial t^\alpha \partial \overline t ^\beta }}.$$


Bounded Domain Inverse Matrix Affine Transformation Hermitian Matrix Bergman Kernel 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Bergman, S., Über die Existenz von Repräsentantenbereichen in Theorie der Abbildung durch Paare von Funktionen zweier komplexen Veränderlichen. Math. Ann. 102 (1929), 430–446.CrossRefGoogle Scholar
  2. [2]
    Lu, Qi-keng, On the Kähler manifold of constant unitary curvature, Acta Mathematica Sinica, 16 (1966), 269–281.MATHGoogle Scholar
  3. [3]
    Skwarzcynski, M., The invariant distance in the theory of pseudoconformal mappings and the Lu Qi-keng conjecture, Proc. Amer. Math. Soc. 22 (1969), 305–310.MathSciNetCrossRefGoogle Scholar
  4. [4]
    Rosenthal, P.L., On the zeros of the Bergman function in doubly-connected domains, Proc. Amer. Math. Soc. 21 (1969), 33–35.MathSciNetMATHCrossRefGoogle Scholar
  5. [5]
    Matsura, S., On the Lu Qi-keng conjecture and the Bergman representative domains, Pacific J. Math., 49 (1973), 407–16.MathSciNetGoogle Scholar
  6. [6]
    Suita, N., and Yamada, A., On the Lu Qi-keng conjecture, Proc. Amer. Math. Soc. 59 (1976), 222–224.MathSciNetMATHCrossRefGoogle Scholar
  7. [7]
    Xu Yichao, The canonical realization of complex homogeneous bounded domains (preprint).Google Scholar

Copyright information

© Birkhäuser Boston, Inc. 1984

Authors and Affiliations

  • Qi-keng Lu
    • 1
  1. 1.Institute of MathematicsAcademia SinicaChina

Personalised recommendations