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

Abstract. Let Ω be a bounded domain in Cn and bΩ is smooth pseudoconvex near z0 ∈ bΩ of finite type. Then there are constants c> 0 and ′> 0 such that the Kobayashi metric, KΩ(z;X), satisfies KΩ(z;X) ≥ c|X|δ(z)− ′ for all X ∈ T 1,0z Cn in a neighborhood of z0. Here δ(z) denotes the distance from z to bΩ. As an application, we prove the Hölder continuity of proper holomorphic maps onto pseudoconvex domains. 1. Introduction. Let Ω ⊂ Cn be a bounded domain in Cn. The purpose of this paper is to study the boundary behavior of the Kobayashi metric, KΩ(z;X), for z near a point z0 ∈ bΩ of finite type. Here finite type means finite 1-type in D’Angelo sense. We will discuss the definition of finite type in section 2. Let us remind the reader of th