A LOWER BOUND ON THE KOBAYASHI METRIC NEAR A POINT OF FINITE TYPE IN C n . Sanghyun Cho Abstract. Let Ω be a bounded domain in C n and bΩ is smooth pseudoconvex near z 0 ∈ bΩ of finite type. Then there are constants c> 0 and ǫ ′ > 0 such that the Kobayashi metric, K Ω (z; X), satisfies K Ω (z; X) ≥ c|X|δ(z) −ǫ ′ for all X ∈ T 1,0 z C n in a neighborhood of z 0 . Here δ(z) denotes the distance from z to bΩ. As an application, we prove the H¨older continuity of proper holomorphic maps onto pseudoconvex domains. 1. Introduction. Let Ω ⊂ C n be a bounded domain in C n . The purpose of this paper is to study the boundary behavior of the Kobayashi metric, K Ω (z ; X ), for z near a point z 0 ∈ 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 the definition of Kobayashi metric. The function K Ω : T 1,0 Ω → R on the holomorphic tangent bundle, given by K Ω (z ; X ) = inf{α> 0; ∃f : △→ Ω holomorphic with f (0) = z, f ′ (0) = α −1 X } = inf{r −1 ; ∃f : △ r → Ω holomorphic with f (0) = z, f ′ (0) = X }, is called the Kobayashi metric of Ω. (Here △ denotes the unit disc and △ r = {t; |t| <r} in C). For a fixed tangent vector X , we will show that K Ω (z ; X ) goes to infinity as z approaches z 0 . Our main result is Theorem 1. Let Ω be a bounded domain in C n and let bΩ is smooth pseudoconvex in a neighborhood U of z 0 ∈ bΩ of finite type. Then there exist a neighborhood V ⊂ U of z 0 and constants c> 0, ǫ ′ > 0 so that for all z ∈ Ω ∩ V and X ∈ T 1,0 z C n K Ω (z ; X ) ≥ c|X |· δ (z ) −ǫ ′ where δ (z ) denotes the distance from z to bΩ. Remark. The exponent ǫ ′ in this theorem will not be the largest possible one. As an application of Theorem 1, we can prove the H¨older continuity for a class of proper holomorphic maps. Let Ω 1 , Ω 2 ⊂⊂ C n be bounded pseudoconvex do- mains in C n and Φ : Ω 1 → Ω 2 be a proper holomorphic map. When Ω 1 satisfies condition R, then the C ∞ -extendability of Φ holds [2,11]. If bΩ 1 is of finite type, then this is the case. Then the question is whether Φ can be extended smoothly up to bΩ 1 with information about Ω 2 . When Ω 1 , Ω 2 are pseudoconvex domains, Henkin has shown that the H¨older continuity of Φ up to Ω 1 can be proved by using the boundary behavior of Kobayashi metric near bΩ 2 [12]. In other words, if the infinitesimal Kobayashi metric on Ω 2 grows sufficiently fast near the boundary of Ω 2 (i.e., K Ω 2 (z ; X ) ≥|X |d(z,bΩ 2 ) −ǫ for some ǫ ∈ (0, 1)), then every proper holo- Typeset by A M S-T E X 1