Carathéodory metric

Summary

The Carathéodory metric, or Carathéodory pseudodistance, of a complex manifold $X$ uses holomorphic maps from $X$ to the unit disc. The distance of two points is the largest Poincaré distance between their images under such maps. It is the smallest pseudodistance for which all holomorphic maps from $X$ to the disc are distance-decreasing, and it is a biholomorphic invariant. It is always at most the Kobayashi metric, and on bounded convex domains the two agree by Lempert's theorem. Constantin Carathéodory introduced it in the 1920s for domains in $\C^n$.