The Kobayashi indicatrix at the center of a circular domain

Abstract

The indicatrix of the Kobayashi infinitesimal metric at the center of a pseudoconvex complete circular domain coincides with this domain. It follows that a nonconvex complete circular domain cannot be biholomorphic to any convex domain. An example shows that a bounded pseudoconvex complete circular domain in C 2 {{\mathbf {C}}^2} need not be taut.