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According to Wikipedia, Schwarz–Pick theorem says that that a holomorphic map of the unit disk into itself decreases the distance of points in the Poincaré metric, which also means that it decreases distances of points for a two-dimensional hyperbolic space under Poincaré disk model.

My question is: Is there an analog of this theorem for three or higher dimensional hyperbolic space, probably under higher-dimensional Poincaré disk model?

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    – Pedro Tamaroff
    Jun 4 '21 at 8:51
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There exists an analogue for holomorphic map from the unit ball in $\bf C^n$ to itself, or even every bounded open set in $\bf C^n$.

In fact every complex manifold $M$ (for instance open set in $C^n$) such that every map $\bf C\to M$ is constant (for instance bonded open set) admit a natural distance called the Kobayashi distance which is decreased by holomorphic maps.

See https://en.wikipedia.org/wiki/Kobayashi_metric

The Kobayashi distance of the ball in $\bf C^n$ is the natural generalization of the Poincaré metric :its isometry group is $SU(n,1)$.

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