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Quantitative localization and comparison of invariant distances of domains in $\mathbb C^n$

Nikolai Nikolov, Pascal J. Thomas

Abstract

We obtain explicit bounds on the difference and ratio between "local" and "global" Kobayashi distances in a domain of $\mathbb C^n$ as the points go toward a boundary point with appropriate geometric properties. We use this for the global comparison of various invariant distances. We provide some sharp estimates in dimension $1$.

Quantitative localization and comparison of invariant distances of domains in $\mathbb C^n$

Abstract

We obtain explicit bounds on the difference and ratio between "local" and "global" Kobayashi distances in a domain of as the points go toward a boundary point with appropriate geometric properties. We use this for the global comparison of various invariant distances. We provide some sharp estimates in dimension .

Paper Structure

This paper contains 15 sections, 21 theorems, 110 equations.

Table of Contents

  1. Introduction
  2. Previous results and motivations.
  3. Some definitions and notations.
  4. Results.
  5. Proofs of Theorem \ref{['kobloc']} and \ref{['finsloc']}
  6. Proof of Theorem \ref{['kobloc']}.
  7. Kobayashi $\varepsilon$-extremals cannot go far
  8. Proof of \ref{['distupest']}.
  9. Proof of Theorem \ref{['finsloc']}, \ref{['distlowest']}.
  10. Proof of Theorem \ref{['finsloc']}, \ref{['uprat']} and \ref{['lowrat']}.
  11. Proofs of Theorems \ref{['berg']}, \ref{['lemp']}, and \ref{['lempcara']}
  12. Proof of Theorem \ref{['berg']}.
  13. Proof of Theorem \ref{['lemp']}.
  14. Proof of Theorem \ref{['lempcara']}.
  15. The planar case

Key Result

Theorem 1.1

Let $D \subset \mathbb C^n$ be a domain. Assume that $D$ is $\mathbb C$-convexifiable near $p \in \partial D$, and that $p$ is of type at most $m.$ Then there exists a neighborhood $U_0$ of $p$ such that for any neighborhoods $V\subset \subset U\subset U_0$, with $D_U$ connected, one may find $C>0$

Theorems & Definitions (37)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • proof
  • Lemma 2.1
  • Proposition 2.2
  • ...and 27 more