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<p>In this paper, we study geometric properties of quasihyperbolic quasigeodesics in domains of real Banach spaces from Gromov hyperbolicity. Our contribution is twofold. Firstly, we establish a necessary and sufficient condition for domains in which quasihyperbolic quasigeodesics are double cone curves. Secondly, using this criterion, we demonstrate that quasihyperbolic quasigeodesics in Gromov hyperbolic John domains are double cone curves. This result provides a partial answer in the affirmative to an open question arisen by Väisälä in the free quasiworld. As applications, we investigate geometric properties of quasihyperbolic quasigeodesics in uniform domains. Moreover, we prove another result which corresponds to the theorem of Väisälä–Heinonen–Näkki regarding distortion properties of quasihyperbolic quasigeodesics under freely quasiconformal mappings in Banach spaces.</p>

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Gromov hyperbolicity in the free quasiworld. II

  • 期刊论文

  • 发表日期：

    2025年8月8日

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  • Qingshan Zhou,
  • Saminathan Ponnusamy,
  • Qianghua Luo

摘要

In this paper, we study geometric properties of quasihyperbolic quasigeodesics in domains of real Banach spaces from Gromov hyperbolicity. Our contribution is twofold. Firstly, we establish a necessary and sufficient condition for domains in which quasihyperbolic quasigeodesics are double cone curves. Secondly, using this criterion, we demonstrate that quasihyperbolic quasigeodesics in Gromov hyperbolic John domains are double cone curves. This result provides a partial answer in the affirmative to an open question arisen by Väisälä in the free quasiworld. As applications, we investigate geometric properties of quasihyperbolic quasigeodesics in uniform domains. Moreover, we prove another result which corresponds to the theorem of Väisälä–Heinonen–Näkki regarding distortion properties of quasihyperbolic quasigeodesics under freely quasiconformal mappings in Banach spaces.