On the Representative Domain

Abstract

Professor Stefan Bergman [1] introduced the idea of Repräsentantenbereich of a bounded domain 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() 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 . Here we use the summation convention and denote by ${\text{k}}({\text{z}},\bar t)$ the Bergman kernel of and by $T^{\overline \beta \alpha } (t,\overline t )$ the elements of the inverse matrix of the Bergman metric tensor

$$ T_{\alpha \overline \beta } (t,\overline t ) = \frac{{\partial ^2 \log \,K(t,\overline t )}}{{\partial t^\alpha \partial \overline t ^\beta }}.$$