About the Caratheodory Completeness of all Reinhardt Domains

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In the theory of complex analysis, there are different notions of distances on a bounded domain, for example, the Caratheodory-distance dealing with bounded holomorphic functions, the Bergmann-metric measuring how many L2-holomorphic functions exist, or the Kobayashi-distance, describing the sizes of analytic discs in G. The main problem working with these distances is to decide the domain G that is complete with respect to one of the distances. Any pseudo-convex domain with C1-boundary is complete with respect to the Bergmann-metric. The Caratheodory-distance can be compared with the other two, in fact, it is the smallest one, but there is no relation between the Bergmann-metric and the Kobayashi-metric. Any bounded complete Reinhardt domain G that is pseudo-convex is complete in the sense of the Caratheodory-distance; in fact, it is proved that any Caratheodory ball is a relatively compact subset of G.

References (8)

  • D. Catlin

    Boundary behaviour of holomorphic functions on pseudoconvex domains

    Journal Diff. Geometry

    (1980)
  • K. Diederich et al.

    Comparison of the Bergmann and the Kobayashi metric

    Math. Annalen

    (1980)
  • T. Franzoni et al.

    Holomorphic and invariant distances

    Notas de Matemática

    (1980)
  • T.W. Gamelin

    Peak points for algebras on circled sets

    Math. Annalen

    (1978)
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