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Holomorphic maps acting as Kobayashi isometries on a family of geodesics

Filippo Bracci, Łukasz Kosiński, Włodzimierz Zwonek

TL;DR

This work characterizes when a holomorphic map $F:D\to G$ that is an isometry for the Kobayashi distance on a complete family of geodesics must be biholomorphic, proving two sharp regimes: (i) $D$ complete hyperbolic with geodesics from a fixed point, and (ii) $D,G$ $C^{2+\alpha}$-smooth bounded strongly pseudoconvex with geodesic rays landing at a boundary point. The authors deploy the scaling method to compare with the unit ball, invoke persistence and convergence of geodesics, and use a combination of properness, differential nondegeneracy, and degree arguments to deduce biholomorphism. They also demonstrate the optimality of the assumptions via holomorphic coverings and Reinhardt-domain constructions, where Kobayashi-isometric behavior along a complete geodesic family does not imply biholomorphism. The results illuminate the limits of rigidity for Kobayashi-isometries and provide concrete model arguments that connect internal and boundary geodesic geometry to holomorphic equivalence. These findings have implications for understanding when geometric constraints on invariant metrics determine complex-analytic structure.

Abstract

Consider a holomorphic map $F: D \to G$ between two domains in ${\mathbb C}^N$. Let $\mathcal F$ denote a family of geodesics for the Kobayashi distance, such that $F$ acts as an isometry on each element of $\mathcal F$. This paper is dedicated to characterizing the scenarios in which the aforementioned condition implies that $F$ is a biholomorphism. Specifically, we establish this when $D$ is a complete hyperbolic domain, and $\mathcal F$ comprises all geodesic segments originating from a specific point. Another case is when $D$ and $G$ are $C^{2+α}$-smooth bounded pseudoconvex domains, and $\mathcal F$ consists of all geodesic rays converging at a designated boundary point of $D$. Furthermore, we provide examples to demonstrate that these assumptions are essentially optimal.

Holomorphic maps acting as Kobayashi isometries on a family of geodesics

TL;DR

This work characterizes when a holomorphic map that is an isometry for the Kobayashi distance on a complete family of geodesics must be biholomorphic, proving two sharp regimes: (i) complete hyperbolic with geodesics from a fixed point, and (ii) -smooth bounded strongly pseudoconvex with geodesic rays landing at a boundary point. The authors deploy the scaling method to compare with the unit ball, invoke persistence and convergence of geodesics, and use a combination of properness, differential nondegeneracy, and degree arguments to deduce biholomorphism. They also demonstrate the optimality of the assumptions via holomorphic coverings and Reinhardt-domain constructions, where Kobayashi-isometric behavior along a complete geodesic family does not imply biholomorphism. The results illuminate the limits of rigidity for Kobayashi-isometries and provide concrete model arguments that connect internal and boundary geodesic geometry to holomorphic equivalence. These findings have implications for understanding when geometric constraints on invariant metrics determine complex-analytic structure.

Abstract

Consider a holomorphic map between two domains in . Let denote a family of geodesics for the Kobayashi distance, such that acts as an isometry on each element of . This paper is dedicated to characterizing the scenarios in which the aforementioned condition implies that is a biholomorphism. Specifically, we establish this when is a complete hyperbolic domain, and comprises all geodesic segments originating from a specific point. Another case is when and are -smooth bounded pseudoconvex domains, and consists of all geodesic rays converging at a designated boundary point of . Furthermore, we provide examples to demonstrate that these assumptions are essentially optimal.
Paper Structure (6 sections, 11 theorems, 45 equations)

This paper contains 6 sections, 11 theorems, 45 equations.

Table of Contents

  1. Introduction
  2. Geodesics in strongly pseudoconvex domains
  3. Application of the scaling method
  4. The proof of Theorem \ref{['main-intro-inner']} and Theorem \ref{['main-intro']}
  5. Special cases: holomorphic coverings and Reinhardt domains
  6. Examples in Reinhardt domains

Key Result

Theorem 1.1

Let $D\subset \mathbb C^N$, $N\geq 1$, be a complete hyperbolic domain and let $G$ be a domain in $\mathbb C^N$. Let $p\in D$ and let $\mathcal{F}$ be the family of all geodesic segments in $D$ starting from $p$. Let $F:D\to G$ be holomorphic. Then $F$ is a biholomorphism if and only if $F\circ \gam

Theorems & Definitions (23)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1
  • proof
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • proof
  • Lemma 2.5
  • ...and 13 more