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The Bergman-Fridman invariant on some classes of pseudoconvex domains

Rahul Kumar, Prachi Mahajan

Abstract

We study the boundary behaviour of a variant of the Fridman's invariant function (defined in terms of the Bergman metric) on Levi corank one domains, strongly pseudoconvex domains, smoothly bounded convex domains in $ \mathbb{C}^n $ and polyhedral domains in $ \mathbb{C}^2 $.

The Bergman-Fridman invariant on some classes of pseudoconvex domains

Abstract

We study the boundary behaviour of a variant of the Fridman's invariant function (defined in terms of the Bergman metric) on Levi corank one domains, strongly pseudoconvex domains, smoothly bounded convex domains in and polyhedral domains in .
Paper Structure (14 sections, 14 theorems, 142 equations)

This paper contains 14 sections, 14 theorems, 142 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Some definitions
  4. Some notation:
  5. Some remarks on the Bergman-Fridman invariant:
  6. Localisation Result:
  7. $h$-extendible domains
  8. Convergence of the Bergman distance on the scaled domains
  9. Proof of Theorem \ref{['T0']}(i) - Levi corank one domain $D$
  10. Completeness of $(D_{\infty}, d^b_{D_{\infty}})$:
  11. Proof of Theorem \ref{['T0']}(ii) - Strongly pseudoconvex domain $D$
  12. Proof of Theorem \ref{['T0']}(iii) - Convex domain $D$
  13. Proof of Theoerem \ref{['T0']} (iv) - Strongly polyhedral domain $D$:
  14. Detecting Strong pseudoconvexity

Key Result

Theorem 1.1

Let $D$ be a bounded domain in $\mathbb{C}^n$ and $\{p^j\}$ be a sequence of points in $D$ that converges to $p^0 \in \partial D$.

Theorems & Definitions (20)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1
  • proof
  • Theorem 2.2
  • Lemma 3.1
  • proof
  • Theorem 4.1
  • ...and 10 more