# On the Representative Domain

• Qi-keng Lu
Chapter

## Abstract

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:

$${\text{z}} \to \frac{{\partial \log \frac{{{\text{k}}({\text{z}},\bar t)}} {{{\text{k}}({\text{t}},\bar t)}}}} {{\partial \bar t^\beta }}{\text{T}}^{\bar \beta \alpha } ({\text{t}},\bar t)$$
(1)
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
$$T_{\alpha \overline \beta } (t,\overline t ) = \frac{{\partial ^2 \log \,K(t,\overline t )}}{{\partial t^\alpha \partial \overline t ^\beta }}.$$

## Keywords

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.

## References

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