Proper holomorphic map

Summary

A proper holomorphic map is a holomorphic map between domains, or complex manifolds, that is also a proper map: preimages of compact sets are compact. For bounded domains this means that the map sends points near the boundary of the source to points near the boundary of the target. Proper holomorphic maps between domains of the same dimension are finite branched coverings, so they generalize biholomorphisms. In several variables they are rigid: by Alexander's theorem, every proper holomorphic self-map of the ball in $\C^n$ with $n \ge 2$ is an automorphism.