A lower bound on the Kobayashi metric near a point of finite type in ℂ n

Abstract

Let Ω be a bounded domain in ℂ n andbΩ smooth pseudoconvex near z0 ∈bΩ of finite type. Then there are constantsc>0 and e′>0 such that the Kobayashi metric,K Ω(z; X), satisfiesK Ω(z; X)≥c|X|δ(z)−t′ for allX ∈T 1,0 ℂ n in a neighborhood ofz 0. Here δ(z) denotes the distance fromz tobΩ. As an application, we prove the Holder continuity of proper holomorphic maps onto pseudoconvex domains.