Estimates of Invariant Metrics on Pseudoconvex Domains of Finite Type inC3

Sanghyun Cho,Young Hwan You

doi:10.1155/2014/697160

Sanghyun Cho, Young Hwan You

Open Access

https://doi.org/10.1155/2014/697160

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Abstract

LetΩbe a smoothly bounded pseudoconvex domain inC3and assume thatz0∈bΩis a point of finite 1-type in the sense of D’Angelo. Then, there are an admissible curveΓ⊂Ω∪{z0}, connecting points q0∈Ωandz0∈bΩ, and a quantityM(z,X), alongz∈Γ, which bounds from above and below the Bergman, Caratheodory, and Kobayashi metrics in a small constant and large constant sense.

Highlights

Let Ω be a smoothly bounded domain in Cn and let X be a holomorphic tangent vector at a point z in Ω, and let us denote the Bergman, Caratheodory, and Kobayashi metrics at Ω z by is a BΩ(z; X), stronglyCΩ(z; X), and pseudoconvexKdΩo(mz;aXin),irnesCpen,cttihveelyo.pWtimheanl boundary behavior of the above metrics is well understood
For weakly pseudoconvex domains of finite type in Cn, several authors found some results about these metrics
To estimate the above invariant metrics, we need a complete geometric analysis near z0 ∈ bΩ of finite type, and we construct a family of plurisubharmonic functions with maximal Hessian near bΩ

Summary

Introduction

Let Ω be a smoothly bounded domain in Cn and let X be a holomorphic tangent vector at a point z in Ω, and let us denote the Bergman, Caratheodory, and Kobayashi metrics at Ω z by is a BΩ(z; X), strongly. To estimate the above invariant metrics, we need a complete geometric analysis near z0 ∈ bΩ of finite type, and we construct a family of plurisubharmonic functions with maximal Hessian near bΩ. Functions z = (z1, z2, z3) defined in a neighborhood V of z0 such that z0 = 0 and |∂r/∂z3| ≥ c0 on V for a uniform constant c0 > 0, and |r(z1, 0, 0)| vanishes to order η, and (∂r/∂z2)(0) = 0 (Theorem 2.1 in [13]) Let Ω ⊂⊂ C3 be a smoothly bounded pseudoconvex domain and assume z0 ∈ bΩ is a point of finite 1-type in the sense of D’Angelo; that is, Δ 1(z0) < ∞. To avoid the difficulty to push out the domain in z1-direction, we use a bumping theorem of Cho [14]

Special Coordinates

Metric Estimates