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The limiting behavior of the Kobayashi-Royden pseudometric


Author: Shulim Kaliman
Journal: Trans. Amer. Math. Soc. 339 (1993), 361-371
MSC: Primary 32H15; Secondary 30F45, 32E10, 32G10
DOI: https://doi.org/10.1090/S0002-9947-1993-1118826-9
MathSciNet review: 1118826
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Abstract: We study the limit of the sequence of Kobayashi metrics of Riemann surfaces (when these Riemann surfaces form an analytic fibration in such a way that the total space of fibration becomes a complex surface), as the fibers approach the center fiber which is not in general smooth. We prove that if the total space is a Stein surface and the smooth part of the center fiber contains a component biholomorphic to a quotient of the disk by a Fuchsian group of first kind, then the Kobayashi metrics of the near-by fibers converge to the Kobayashi metric of this component as fibers tend to the center fiber.


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  • [A] W. Abikoff, The real analytic theory of Teichmüller space, Lecture Notes in Math., vol. 820, Springer-Verlag, Berlin and New York, 1980. MR 590044 (82a:32028)
  • [B] L. Bers, Uniformization, moduli, and Kleinian groups, Bull. London Math. Soc. 4 (1972), 257-300. MR 0348097 (50:595)
  • [G] G. M. Goluzin, Geometric theory of functions of a complex variable, Amer. Math. Soc., Providence, R.I., 1969. MR 0247039 (40:308)
  • [GR] R. G. Gunning and H. Rossi, Analytic functions of several complex variables, Prentice-Hall, Englewood Cliffs, N.J., 1965. MR 0180696 (31:4927)
  • [Ka] Sh. Kaliman, Polynomials on $ {C^2}$ with isomorphic generic fibers, Soviet Math. Dokl. 33 (1986), 600-603.
  • [Ko] S. Kobayashi, Hyperbolic manifolds and holomorphic mappings, Marcel Dekker, New York, 1970. MR 0277770 (43:3503)
  • [PS] E. A. Poletsky and B. V. Shabat, Invariant metrics, Itogi Nauki i Tekhniki (Modern Problems of Mathematics, Fundamental Directions), VINITI, Moscow, 1986, pp. 73-125. (Russian) MR 860610
  • [R] H. Royden, Remarks on Kobayashi metric, Several Complex Variables II (Proc. Internat. Conf., College Park, Md., 1970), Lecture Notes in Math., vol. 185, Springer-Verlag, 1971, pp. 125-137. MR 0304694 (46:3826)
  • [S] Y.-T. Siu, Every Stein subvariety admits a Stein neighborhood, Invent. Math. 38 (1976), 89-100. MR 0435447 (55:8407)
  • [Z] M. G. Zaidenberg, On hyperbolic embeddings of complements of divisors and the limiting behavior of the Kobayashi-Royden metric, Math. USSR-Sb. 55 (1986), 55-70. MR 791317 (86k:32021)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1993-1118826-9
Article copyright: © Copyright 1993 American Mathematical Society

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