Holomorphic maps which preserve intrinsic metrics or measures

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.