Circular Models and Normal Forms for Convex Domains

Abstract

In [L1], Lempert studied the properties of the infinitesimal Kobayashi metric on smoothly bounded strongly convex domains in ℂ n . He showed that the exponential map for the infinitesimal Kobayashi metric (which is a Finsler metric) is a smooth diffeomorphism from the tangent space minus the origin onto the domain minus the base point; moreover, if the map is suitably renormalized, then the map restricts to any complex line through the origin as a biholomorphic map from a unit (Kobayashi) disc in the tangent line onto a proper holomorphic curve in the domain. He also realized that this map could be used for the analysis of the equivalence classes of pointed domains. In [L2], he discussed normal forms for domains along the boundary of extremal discs, and produced analytic modular data for the class of pointed framed convex domains.