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Worm Domains are not Gromov Hyperbolic.

Leandro Arosio1, Gian Maria Dall'Ara2, Matteo Fiacchi3

  • 1Department of Mathematics, University of Rome"Tor Vergata", Rome, Italy.

Journal of Geometric Analysis
|June 5, 2023
PubMed
Summary

Worm domains are not Gromov hyperbolic. This study demonstrates that these specific mathematical domains lack Gromov hyperbolicity when measured by the Kobayashi distance.

Keywords:
Gromov hyperbolicityKobayashi metricScalingWorm domain

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Area of Science:

  • Complex analysis
  • Geometric measure theory

Background:

  • Gromov hyperbolicity is a key concept in geometric group theory and complex analysis.
  • Kobayashi distance is a metric used to measure distances in complex manifolds.

Purpose of the Study:

  • To investigate the Gromov hyperbolicity of Worm domains.
  • To determine if Worm domains satisfy the criteria for Gromov hyperbolicity under the Kobayashi distance.

Main Methods:

  • Analysis of the geometric properties of Worm domains.
  • Application of the definition of Gromov hyperbolicity using the Kobayashi distance.

Main Results:

  • Worm domains are demonstrated to not be Gromov hyperbolic.
  • The Kobayashi distance on Worm domains does not satisfy the conditions required for Gromov hyperbolicity.

Conclusions:

  • Worm domains lack Gromov hyperbolicity with respect to the Kobayashi distance.
  • This finding has implications for the study of complex manifolds and their geometric properties.