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On the Approaching Geodesic Property via the Quotient Invariant

Kingshook Biswas, Sanjoy Chatterjee

TL;DR

The paper links the approaching geodesic property to the quotient invariant q_D on Kobayashi hyperbolic domains in C^d, showing that q_D(z) → 1 toward a boundary point implies the domain has approaching geodesics at that point. The authors extend squeezing-function-based results and prove the main theorem via a ball-automorphism limit argument, Montel/Arzelà–Ascoli, and Kobayashi distance non-expansiveness. Consequences include a visibility property and a homeomorphism between horofunction, Gromov, and Euclidean closures, especially for convex domains. These results refine the understanding of the boundary geometry of Kobayashi hyperbolic domains and their compactifications.

Abstract

We study the approaching geodesic property of a bounded domain in $\mathbb{C}^{n}$ with respect to the Kobayashi distance using the quotient invariant.

On the Approaching Geodesic Property via the Quotient Invariant

TL;DR

The paper links the approaching geodesic property to the quotient invariant q_D on Kobayashi hyperbolic domains in C^d, showing that q_D(z) → 1 toward a boundary point implies the domain has approaching geodesics at that point. The authors extend squeezing-function-based results and prove the main theorem via a ball-automorphism limit argument, Montel/Arzelà–Ascoli, and Kobayashi distance non-expansiveness. Consequences include a visibility property and a homeomorphism between horofunction, Gromov, and Euclidean closures, especially for convex domains. These results refine the understanding of the boundary geometry of Kobayashi hyperbolic domains and their compactifications.

Abstract

We study the approaching geodesic property of a bounded domain in with respect to the Kobayashi distance using the quotient invariant.

Paper Structure

This paper contains 4 sections, 8 theorems, 33 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and Technical Results
  3. Proof of Theorem \ref{['T:approachsgeo']}
  4. Consequences of approaching geodesics

Key Result

Theorem 1.6

Let $D \Subset \mathbb{C}^d$ be a complete hyperbolic domain and $\zeta \in \partial_{H}D$. Suppose that $q_{D}(z) \to 1$ as $z \to \zeta$. Then the domain $D$ has the approaching geodesic property at $\zeta$.

Theorems & Definitions (29)

  • Definition 1.4
  • Remark 1.5
  • Theorem 1.6
  • Remark 1.7
  • Theorem 1.8
  • Definition 2.1: Gromov compactification
  • Definition 2.2: Horofunction compactifiation
  • Definition 2.3
  • Definition 2.4
  • Lemma 3.1
  • ...and 19 more