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Weighted boundary limits of the Kobayashi--Fuks metric on h-extendible domains

Debaprasanna Kar

Abstract

We study the boundary behavior of the Kobayashi--Fuks metric on the class of h-extendible domains. Here, we derive the non-tangential boundary asymptotics of the Kobayashi--Fuks metric and its Riemannian volume element by the help of some maximal domain functions and then using their stability results on h-extendible local models.

Weighted boundary limits of the Kobayashi--Fuks metric on h-extendible domains

Abstract

We study the boundary behavior of the Kobayashi--Fuks metric on the class of h-extendible domains. Here, we derive the non-tangential boundary asymptotics of the Kobayashi--Fuks metric and its Riemannian volume element by the help of some maximal domain functions and then using their stability results on h-extendible local models.
Paper Structure (4 sections, 8 theorems, 67 equations)

This paper contains 4 sections, 8 theorems, 67 equations.

Table of Contents

  1. Introduction
  2. Notations, Definitions and Main Results
  3. Maximal domain functions and stability
  4. Proofs of the results

Key Result

Theorem 2.2

Let $\Omega$ be a bounded pseudoconvex domain in $\mathbb C^n$ and $p\in \partial \Omega$ an h-extendible boundary point with Catlin's multitype $(m_1,\ldots,m_n)$. Then there are local coordinates $z$, and a local model $D_0$ at $p$ such that Here $\Gamma$ is a non-tangential cone in $\Omega$ with vertex at $p$, $\pi_{1/d(z)}(u)=(d(z)^{-1/m_1}u_1,\ldots,d(z)^{-1/m_n}u_n)$, the unit vector $u^*=\

Theorems & Definitions (13)

  • Definition 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Corollary 2.4
  • Corollary 2.5
  • Lemma 3.1
  • Proposition 3.2
  • Lemma 3.3: Localization
  • proof : Proof of Theorem \ref{['kf metric']}
  • Lemma 4.1
  • ...and 3 more