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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Invariant distances related to the Bergman function


Authors: T. Mazur, P. Pflug and M. Skwarczyński
Journal: Proc. Amer. Math. Soc. 94 (1985), 72-76
MSC: Primary 32H15; Secondary 32H10
MathSciNet review: 781059
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Abstract: Let $ D$ be a bounded domain in $ {{\mathbf{C}}^n}$. The invariant distance in $ D$ is given by

$\displaystyle {\rho _D}(z,w) = {\left( {1 - {{\left( {\frac{{{K_D}(z,w){K_D}(w,z)}}{{{K_D}(z,z){K_D}(w,w)}}} \right)}^{1/2}}} \right)^{1/2}}.$

It is shown that one half of the length of a piecewise $ {C^1}$ curve $ \gamma :[a,b] \to D$ with respect to the Bergman metric is equal to the length of $ \gamma $ measured by $ {\rho _D}$, which implies that the associated inner distance $ \rho _D^*$ coincides (up to the factor $ \tfrac{1}{2}$) with the Bergman-distance. Also it was proved that $ {\rho _D}$ is not an inner distance.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1985-0781059-0
PII: S 0002-9939(1985)0781059-0
Article copyright: © Copyright 1985 American Mathematical Society