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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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The limiting behavior of the Kobayashi-Royden pseudometric
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by Shulim Kaliman PDF
Trans. Amer. Math. Soc. 339 (1993), 361-371 Request permission


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.
  • William Abikoff, The real analytic theory of Teichmüller space, Lecture Notes in Mathematics, vol. 820, Springer, Berlin, 1980. MR 590044
  • Lipman Bers, Uniformization, moduli, and Kleinian groups, Bull. London Math. Soc. 4 (1972), 257–300. MR 348097, DOI 10.1112/blms/4.3.257
  • G. M. Goluzin, Geometric theory of functions of a complex variable, Translations of Mathematical Monographs, Vol. 26, American Mathematical Society, Providence, R.I., 1969. MR 0247039
  • Robert C. Gunning and Hugo Rossi, Analytic functions of several complex variables, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1965. MR 0180696
  • Sh. Kaliman, Polynomials on ${C^2}$ with isomorphic generic fibers, Soviet Math. Dokl. 33 (1986), 600-603.
  • Shoshichi Kobayashi, Hyperbolic manifolds and holomorphic mappings, Pure and Applied Mathematics, vol. 2, Marcel Dekker, Inc., New York, 1970. MR 0277770
  • E. A. Poletskiĭ and B. V. Shabat, Invariant metrics, Current problems in mathematics. Fundamental directions, Vol. 9 (Russian), Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1986, pp. 73–125, 292 (Russian). MR 860610
  • H. L. Royden, Remarks on the Kobayashi metric, Several complex variables, II (Proc. Internat. Conf., Univ. Maryland, College Park, Md., 1970) Lecture Notes in Math., Vol. 185, Springer, Berlin, 1971, pp. 125–137. MR 0304694
  • Yum Tong Siu, Every Stein subvariety admits a Stein neighborhood, Invent. Math. 38 (1976/77), no. 1, 89–100. MR 435447, DOI 10.1007/BF01390170
  • M. G. Zaĭdenberg, Hyperbolic imbedding of complements to divisors and the limit behavior of the Kobayashi-Royden metric, Mat. Sb. (N.S.) 127(169) (1985), no. 1, 55–71, 143 (Russian). MR 791317
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Additional Information
  • © Copyright 1993 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 339 (1993), 361-371
  • MSC: Primary 32H15; Secondary 30F45, 32E10, 32G10
  • DOI:
  • MathSciNet review: 1118826