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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

Holomorphic maps which preserve intrinsic metrics or measures


Author: Ian Graham
Journal: Trans. Amer. Math. Soc. 319 (1990), 787-803
MSC: Primary 32H15
MathSciNet review: 967313
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Abstract: Suppose that $ M$ is a domain in a taut complex manifold $ M'$, and that $ \Omega $ is a strictly convex bounded domain in $ {{\mathbf{C}}^n}$. We consider the following question: given a holomorphic map $ F:M \to \Omega $ which is an isometry for the infinitesimal Kobayashi metric at one point, must $ F$ be biholomorphic? With an additional technical assumption on the behavior of the Kobayashi distance near points of $ \partial M$, we show that $ F$ gives a biholomorphism of $ M$ with an open dense subset of $ \Omega $. Moreover, $ F$ extends as a homeomorphism from a larger domain $ \tilde M$ to $ \Omega $. We also give some related results--refinements of theorems of Bland and Graham and Fornaess and Sibony, and the answer to a question of Graham and Wu.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1990-0967313-4
PII: S 0002-9947(1990)0967313-4
Article copyright: © Copyright 1990 American Mathematical Society