Holomorphic Mappings into Convex Domains

Abstract

We consider two problems for holomorphic maps into convex domains in ℂn. The first is to give conditions under which a holomorphic map which is an isometry for the infinitesimal Kobayashi metric at one point must be biholomorphic. The second is to estimate the distortion for a holomorphic map into a convex domain in terms of a readily computable quantity, namely the radii of certain linear discs in the domain and target space of the map. (In problem 2 the domain of the mapping is another domain in ℂn.) This leads to estimates for the Kobayashi metric on convex domains, and a version of the Koebe theorem. Sharp constants have now been obtained for these results [G5].KeywordsTarget SpaceConvex DomainUnique ComplexTechnical AssumptionStrong Maximum PrincipleThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.