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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Invariant distances related to the Bergman function
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by T. Mazur, P. Pflug and M. Skwarczyński PDF
Proc. Amer. Math. Soc. 94 (1985), 72-76 Request permission

Abstract:

Let $D$ be a bounded domain in ${{\mathbf {C}}^n}$. The invariant distance in $D$ is given by \[ {\rho _D}(z,w) = {\left ( {1 - {{\left ( {\frac {{{K_D}(z,w){K_D}(w,z)}}{{{K_D}(z,z){K_D}(w,w)}}} \right )}^{1/2}}} \right )^{1/2}}.\] 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
  • © Copyright 1985 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 94 (1985), 72-76
  • MSC: Primary 32H15; Secondary 32H10
  • DOI: https://doi.org/10.1090/S0002-9939-1985-0781059-0
  • MathSciNet review: 781059