Kobayashi metric

Summary

The Kobayashi metric, or Kobayashi pseudodistance, of a complex manifold $X$ is a pseudodistance on $X$ built from holomorphic discs. It is the largest pseudodistance for which every holomorphic map from the unit disc, with its Poincaré distance, is distance-decreasing. It is built from chains of holomorphic discs, and Shoshichi Kobayashi introduced it in 1967. Every holomorphic map between complex manifolds is distance-decreasing for the Kobayashi pseudodistances, so it is a biholomorphic invariant. It vanishes identically on $\C^n$ and is a true distance on bounded domains. The manifolds on which it is a distance are the Kobayashi hyperbolic manifolds.