Journal of Geometric Analysis

, Volume 22, Issue 2, pp 491–560 | Cite as

Some Aspects of the Kobayashi and Carathéodory Metrics on Pseudoconvex Domains

  • Prachi MahajanEmail author
  • Kaushal Verma


The purpose of this article is to consider two themes, both of which emanate from and involve the Kobayashi and the Carathéodory metric. First, we study the biholomorphic invariant introduced by B. Fridman on strongly pseudoconvex domains, on weakly pseudoconvex domains of finite type in C 2, and on convex finite type domains in C n using the scaling method. Applications include an alternate proof of the Wong–Rosay theorem, a characterization of analytic polyhedra with noncompact automorphism group when the orbit accumulates at a singular boundary point, and a description of the Kobayashi balls on weakly pseudoconvex domains of finite type in C 2 and convex finite type domains in C n in terms of Euclidean parameters. Second, a version of Vitushkin’s theorem about the uniform extendability of a compact subgroup of automorphisms of a real analytic strongly pseudoconvex domain is proved for C 1-isometries of the Kobayashi and Carathéodory metrics on a smoothly bounded strongly pseudoconvex domain.


Kobayashi metric Carathéodory metric Fridman’s invariant Scaling Isometry 

Mathematics Subject Classification (2000)

32F45 32Q45 


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© Mathematica Josephina, Inc. 2010

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of ScienceBangaloreIndia

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