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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
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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Article copyright: © Copyright 1993 American Mathematical Society