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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
Article

Abstract

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.

Keywords

Kobayashi metric Carathéodory metric Fridman’s invariant Scaling Isometry 

Mathematics Subject Classification (2000)

32F45 32Q45 

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References

  1. 1.
    Aladro, G.: The comparability of the Kobayashi approach region and the admissible approach region. Ill. J. Math. 33, 42–63 (1989) MathSciNetzbMATHGoogle Scholar
  2. 2.
    Alexander, H., Wermer, J.: Several Complex Variables and Banach Algebras, 3rd edn. Graduate Texts in Mathematics, vol. 35. Springer, New York (1998) zbMATHGoogle Scholar
  3. 3.
    Berteloot, F.: Characterisation of models in C 2 by their automorphism groups. Int. J. Math. 5, 619–634 (1994) MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Berteloot, F.: Principe de Bloch et estimations de la métrique de Kobayashi des domaines de C 2. J. Geom. Anal. 13, 29–37 (2003) MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Berteloot, F., Coeuré, G.: Domaines de B 2, pseudoconvexes et de type fini ayant un groupe non compact d’automorphismes. Ann. Inst. Fourier 41, 77–86 (1991) zbMATHCrossRefGoogle Scholar
  6. 6.
    Catlin, D.: Estimates of invariant metrics on pseudoconvex domains of dimension two. Math. Z. 200, 429–466 (1989) MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Coupet, B.: Uniform extendibility of automorphisms. Contemp. Math. 137, 177–183 (1992) MathSciNetCrossRefGoogle Scholar
  8. 8.
    Coupet, B., Pinchuk, S.I.: Holomorphic equivalence problem for weighted homogeneous rigid domains in C n+1. In: Complex Analysis in Modern Mathematics, pp. 57–70. FAZIS, Moscow (2001) (Russian) Google Scholar
  9. 9.
    Coupet, B., Sukhov, A.: On the uniform extendibility of proper holomorphic mappings. Complex Var. Theory Appl. 28, 243–248 (1996) MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Coupet, B., Pinchuk, S., Sukhov, A.: On boundary rigidity and regularity of holomorphic mappings. Int. J. Math. 7, 617–643 (1996) MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Coupet, B., Gaussier, H., Sukhov, A.: Regularity of CR maps between convex hypersurfaces of finite type. Proc. Am. Math. Soc. 127, 3191–3200 (1999) MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Fornaess, J.E., Sibony, N.: Increasing sequence of complex manifolds. Math. Ann. 255, 351–360 (1981) MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Forstneric, F., Rosay, J.P.: Localisation of the Kobayashi metric and the boundary continuity of proper holomorphic mappings. Math. Ann. 279, 239–252 (1987) MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Fridman, B.L.: Biholomorphic invariants of a hyperbolic manifold and some applications. Trans. Am. Math. Soc. 276, 685–698 (1983) MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Gaussier, H.: Characterization of convex domains with noncompact automorphism group. Mich. Math. J. 44, 375–388 (1997) MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Graham, I.: Boundary behaviour of the Carathéodory and Kobayashi metrics on strongly pseudoconvex domains in C n with smooth boundary. Trans. Am. Math. Soc. 207, 219–240 (1975) zbMATHGoogle Scholar
  17. 17.
    Greene, R.E., Krantz, S.G.: Deformation of complex structures, estimates for the \(\bar{\partial}\) equation and stability of the Bergman kernel. Adv. Math. 43(1), 1–86 (1982) MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Greene, R.E., Krantz, S.G.: Stability of the Carathéodory and Kobayashi metrics and applications to biholomorphic mappings. In: Complex Analysis of Several Variables, Madison, Wis., 1982. Proc. Sympos. Pure Math., vol. 41, pp. 77–93. Am. Math. Soc., Providence (1984) Google Scholar
  19. 19.
    Herbort, G.: Estimation on invariant distances on the pseudoconvex domains of finite type in dimension two. Math. Z. 251, 673–703 (2005) MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Hoffman, K.: Minimal boundaries for analytic polyhedra. Rend. Circ. Mat. Palermo 9, 147–160 (1960) MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Jarnicki, M., Pflug, P.: Invariant Distances and Metrics in Complex Analysis. de Gruyter Expositions in Mathematics, vol. 9. de Gruyter, Berlin (1993) zbMATHCrossRefGoogle Scholar
  22. 22.
    Kim, K.T., Krantz, S.G.: A Kobayashi metric version of Bun Wong’s Theorem. Complex Var. Elliptic Equ. 54, 355–369 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Kim, K.T., Ma, D.: Characterisation of the Hilbert ball by its automorphisms. J. Korean Math. Soc. 40, 503–516 (2003) MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Kim, K.T., Pagano, A.: Normal analytic polyhedra in C 2 with a noncompact automorphism group. J. Geom. Anal. 11, 283–293 (2001) MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Krantz, S.G.: Invariant metrics and the boundary behavior of holomorphic functions on domains in C n. J. Geom. Anal. 1, 71–97 (1991) MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Lempert, L.: La métrique de Kobayashi et la représentation des domaines sur la boule. Bull. Soc. Math. Fr. 109, 427–474 (1981) MathSciNetzbMATHGoogle Scholar
  27. 27.
    McNeal, J.D.: Convex domains of finite type. J. Funct. Anal. 108, 361–373 (1992) MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    McNeal, J.D.: Estimates on the Bergman kernels of convex domains. Adv. Math. 109, 108–139 (1994) MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Nagel, A., Stein, E.M., Wainger, S.B.: Balls and metrics defined by vector fields I: basic properties. Acta Math. 155, 103–147 (1985) MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Pinchuk, S.I.: Holomorphic inequivalence of certain classes of domains in C n. Math. USSR Sb. 39, 61–86 (1980) CrossRefGoogle Scholar
  31. 31.
    Pinchuk, S.I.: Homogeneous domains with piecewise smooth boundaries. Mat. Zametki 32, 729–735 (1982) MathSciNetGoogle Scholar
  32. 32.
    Rosay, J.-P.: Sur une caractèrisation de la boule parmi les domaines de C n par son groupe d’automorphismes. Ann. Inst. Fourier 29, 91–97 (1979) MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Royden, H.L.: Remarks on the Kobayashi metric. In: Several Complex Variables II. Proc. Internat. Conf., Univ. Maryland, College Park, Md., 1970. Lecture Notes in Math., vol. 185, pp. 125–137. Springer, Berlin (1971) Google Scholar
  34. 34.
    Rudin, W.: Function Theory in the Unit Ball of C n. Grundlehren der Mathematischen Wissenschaften, vol. 241. Springer, New York (1980) CrossRefGoogle Scholar
  35. 35.
    Seshadri, H., Verma, K.: On isometries of the Carathéodory and Kobayashi metrics on strongly pseudoconvex domains. Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) V, 393–417 (2006) MathSciNetGoogle Scholar
  36. 36.
    Seshadri, H., Verma, K.: On the compactness of isometry groups in complex analysis. Complex Var. Elliptic Equ. 54, 387–399 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Venturini, S.: Comparison between the Kobayashi and Carathéodory distances on strongly pseudoconvex bounded domains in C n. Proc. Am. Math. Soc. 107, 725–730 (1989) MathSciNetzbMATHGoogle Scholar
  38. 38.
    Wong, B.: Characterisation of the unit ball in C n by its automorphism group. Invent. Math. 41, 253–257 (1977) MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    Yu, J.: Multitypes of convex domains. Indiana Univ. Math. J. 41, 837–849 (1992) MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    Yu, J.: Weighted boundary limits of the generalized Kobayashi-Royden metrics on weakly pseudoconvex domains. Trans. Am. Math. Soc. 347, 587–614 (1995) zbMATHCrossRefGoogle Scholar

Copyright information

© Mathematica Josephina, Inc. 2010

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of ScienceBangaloreIndia

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