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Gromov Hyperbolicity, John Spaces, and Quasihyperbolic Geodesics

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Abstract

We show that every quasihyperbolic geodesic in a John space admitting a roughly starlike Gromov hyperbolic quasihyperbolization is a double cone arc. This result provides a new approach to an elementary metric geometry question, formulated by Heinonen (Quasiconformal mappings onto John domains. Rev Math Iberoam 5:97–123, 1989), which has been studied by Gehring et al. (Quasihyperbolic geodesics in John domains. Math Scand 36:75–92, 1989). As an application, we obtain a simple geometric condition connecting uniformity of a metric space with the existence of a Gromov hyperbolic quasihyperbolization.

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References

  1. Balogh, Z.M., Buckley, S.M.: Geometric characterizations of Gromov hyperbolicity. Invent. Math. 153, 261–301 (2003)

    Article  MathSciNet  Google Scholar 

  2. Bonk, M., Heinonen, J., Koskela, P.: Uniformizing Gromov hyperbolic domains. Astérisque 270, 1–99 (2001)

    MATH  Google Scholar 

  3. Bridson, M., Haefliger, A.: Metric Spaces of Non-positive Curvature. Springer, Berlin (1999)

    Book  Google Scholar 

  4. Buckley, S., Herron, D.: Uniform spaces and weak slice spaces. Conform. Geom. Dyn. 11, 191–206 (2007). ((electronic))

    Article  MathSciNet  Google Scholar 

  5. Chen, S., Ponnusamy, S.: John disks and K-quasiconformal harmonic mappings. J. Geom. Anal. 27, 1468–1488 (2017)

    Article  MathSciNet  Google Scholar 

  6. Gehring, F.W., Hag, K., Martio, O.: Quasihyperbolic geodesics in John domains. Math. Scand. 36, 75–92 (1989)

    Article  MathSciNet  Google Scholar 

  7. Gehring, F.W., Osgood, B.G.: Uniform domains and the quasi-hyperbolic metric. J. Anal. Math. 36, 50–74 (1979)

    Article  Google Scholar 

  8. Gehring, F.W., Palka, B.P.: Quasiconformally homogeneous domains. J. Anal. Math. 30, 172–199 (1976)

    Article  MathSciNet  Google Scholar 

  9. Ghamsari, M., Näkki, R., Väisälä, J.: John disks and extension of maps. Mon. Math. 117, 63–94 (1994)

    Article  MathSciNet  Google Scholar 

  10. Goldstein, P., Guo, C., Koskela, P., Nandi, D.: Characterizations of generalized John domains i \({R}^ n\) via metric duality. arXiv preprint (2017). arXiv:1710.02050

  11. Guo, C.Y.: Uniform continuity of quasiconformal mappings onto generalized John domains. Ann. Acad. Sci. Fenn. Math. 40, 183–202 (2015)

    Article  MathSciNet  Google Scholar 

  12. Heinonen, J.: Quasiconformal mappings onto John domains. Rev. Math. Iberoam. 5, 97–123 (1989)

    Article  MathSciNet  Google Scholar 

  13. Herron, D.: Geometry and topology of intrinsic distances. J. Anal. 18, 197–231 (2010)

    MathSciNet  MATH  Google Scholar 

  14. Huang, M., Rasila, A., Wang, X., Zhou, Q.: Semisolidity and locally weak quasisymmetry of homeomorphisms in metric spaces. Stud. Math. 242, 267–301 (2018)

    Article  MathSciNet  Google Scholar 

  15. John, F.: Rotation and strain. Commun. Pure Appl. Math. 14, 391–413 (1961)

    Article  MathSciNet  Google Scholar 

  16. Koskela, P., Lammi, P., Manojlović, V.: Gromov hyperbolicity and quasihyperbolic geodesics. Ann. Sci. Éc. Norm. Supér. 47, 975–990 (2014)

    Article  MathSciNet  Google Scholar 

  17. Li, Y.: Near geodesic in John domain in Banach spaces. Int. J. Math. 25(5), 1450041 (2014). https://doi.org/10.1142/S0129167X14500414.

    Article  MATH  Google Scholar 

  18. Li, Y., Vuorinen, M., Zhou, Q.: Characterizations of John spaces. Mon. Math. 188, 547–559 (2019)

    Article  MathSciNet  Google Scholar 

  19. Martio, O., Sarvas, J.: Injectivity theorems in plane and space. Ann. Acad. Sci. Fenn. A 4, 383–401 (1978)

    MathSciNet  MATH  Google Scholar 

  20. Näkki, R., Väisälä, J.: John disks. Expo. Math. 9, 3–43 (1991)

    MathSciNet  MATH  Google Scholar 

  21. Rasila, A., Talponen, J.: On quasihyperbolic geodesics in Banach spaces. Ann. Acad. Sci. Fenn. Math. 39(1), 163–173 (2014)

    Article  MathSciNet  Google Scholar 

  22. Väisälä, J.: Hyperbolic and uniform domains in Banach spaces. Ann. Acad. Sci. Fenn. Math. 30, 261–302 (2005)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors are indebted to the anonymous referee for his/her remarks that helped to improve this manuscript. The first author was supported by NNSF of China (No. 11901090), the Department of Education of Guangdong Province, China (No. 2021KTSCX116), Guangdong–Hong Kong–Macao Intelligent Micro–Nano Optoelectronic Technology Joint Laboratory (Project No. 2020B1212030010), and the Guangdong Basic and Applied Basic Research Foundation (No. 2021A1515110484). The second author was supported by Scientific Research Fund of Hunan Provincial Education Department (No. 20B118) and NSF of Hunan Province (No. 2021JJ30168). The second and the third authors were supported by NNSF of China (No. 11971124) and by NSF of Guangdong Province (No. 2021A1515010326).

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Authors and Affiliations

  1. School of Mathematics and Big Data, Foshan University, Foshan, 528000, Guangdong, People’s Republic of China

    Qingshan Zhou

  2. Department of Mathematics, Hunan First Normal University, Changsha, 410205, Hunan, People’s Republic of China

    Yaxiang Li

  3. Mathematics with Computer Science Program, Guangdong Technion, 241 Daxue Road, Jinping District, Shantou, 515063, Guangdong, People’s Republic of China

    Antti Rasila

  4. Department of Mathematics, Technion - Israel Institute of Technology, 32000, Haifa, Israel

    Antti Rasila

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  1. Qingshan Zhou
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  2. Yaxiang Li
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  3. Antti Rasila
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Correspondence to Yaxiang Li.

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Zhou, Q., Li, Y. & Rasila, A. Gromov Hyperbolicity, John Spaces, and Quasihyperbolic Geodesics. J Geom Anal 32, 228 (2022). https://doi.org/10.1007/s12220-022-00968-2

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