Holomorphic maps which preserve intrinsic metrics or measures

Graham, Ian

doi:10.1090/S0002-9947-1990-0967313-4

Holomorphic maps which preserve intrinsic metrics or measures
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Trans. Amer. Math. Soc. 319 (1990), 787-803 Request permission

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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