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Gromov hyperbolicity of pseudoconvex finite type domains in \({\mathbb {C}}^2\)

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Abstract

We prove that every bounded smooth domain of finite D’Angelo type in \({\mathbb {C}}^2\) endowed with the Kobayashi distance is Gromov hyperbolic and its Gromov boundary is canonically homeomorphic to the Euclidean boundary. We also show that any domain in \({\mathbb {C}}^2\) endowed with the Kobayashi distance is Gromov hyperbolic provided there exists a sequence of automorphisms that converges to a smooth boundary point of finite D’Angelo type.

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References

  1. Abate, M., Patrizio, G.: Finsler Metrics—A Global Approach: with Applications to Geometric Function Theory (1994)

  2. Bedford, E., Pinchuk, S.: Domains in \(\mathbb{C}^{n+1}\) with non-compact automorphism group. J. Geom. Anal. 1, 165–191 (1991)

    Article  MathSciNet  Google Scholar 

  3. Bedford, E., Pinchuk, S.: Domains in \(\mathbb{C}^2\) with noncompact automorphism groups. Indiana Univ. Math. J. 47(1), 199–222 (1998)

    Article  MathSciNet  Google Scholar 

  4. Behan, D.F.: Commuting analytic functions without fixed points. Proc. Am. Math. Soc. 37, 114–120 (1973)

    Article  MathSciNet  Google Scholar 

  5. Bell, S.: Biholomorphic mappings and the \(\overline{\partial }\)-problem. Ann. Math. Sec. Ser. 114(1), 103–13 (1981). https://doi.org/10.2307/1971379

    Article  MathSciNet  MATH  Google Scholar 

  6. Benoist, Y.: Convexes hyperboliques et fonctions quasisymétriques. Publ. Math. Inst. Hautes Etudes Sci. 97, 181–237 (2003)

    Article  Google Scholar 

  7. Berteloot, F.: Characterization of models in \(\mathbb{C}^2\) by their automorphism groups. Int. J. Math. 5, 619–634 (1994)

    Article  Google Scholar 

  8. Berteloot, F.: Méthodes de changement déchelles en analyse complexe. Ann. Faculté Sci. Toulouse Math. 15(3), 427–483 (2006)

    Article  MathSciNet  Google Scholar 

  9. Bharali, G., Zimmer, A.: Goldilocks domains, a weak notion of visibility, and applications. Adv. Math. (2016). https://doi.org/10.1016/j.aim.2017.02.005

    Article  MATH  Google Scholar 

  10. Bracci, F.: Common fixed points of commuting holomorphic maps in the unit ball of \(\mathbb{C}^n\). Proc. Am. Math. Soc. 127, 1133–1141 (1999)

    Article  MathSciNet  Google Scholar 

  11. Bracci, F.: Commuting holomorphic maps in strongly convex domains. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 27(1), 131–144 (1998)

    MathSciNet  MATH  Google Scholar 

  12. Bracci, F., Gaussier, H.: A proof for the Muir–Suffridge conjecture for convex maps of the unit ball of \(\mathbb{C}^n\). Math. Ann. 372(1–2), 845–858 (2018)

    Article  MathSciNet  Google Scholar 

  13. Bracci, F., Gaussier, H.: Horosphere topology. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 20(1), 239–289 (2020)

    MathSciNet  MATH  Google Scholar 

  14. Bracci, F., Gaussier, H., Zimmer, A.: Homeomorphic extension of quasi-isometries for convex domains in \(\mathbb{C}^d\) and iteration theory. Math. Ann. (2020). https://doi.org/10.1007/s00208-020-01954-1

    Article  MATH  Google Scholar 

  15. Bracci, F., Saracco, A.: Hyperbolicity in unbounded convex domains. Forum Math. 5(21), 815–826 (2009)

    MathSciNet  MATH  Google Scholar 

  16. Bridson, M., Haefliger, A.: Metric spaces of non-positive curvature (2009)

  17. Balogh, Z., Bonk, M.: M Gromov hyperbolicity and the Kobayashi metric on strictly pseudoconvex domains. Comment. Math. Helv. 75(3), 504–533 (2000)

    Article  MathSciNet  Google Scholar 

  18. Calamai, S.: A bounded Kohn Nirenberg domain. Bull. Korean Math. Soc. 51, 1339–1345 (2014)

    Article  MathSciNet  Google Scholar 

  19. Catlin, D.W.: Estimates of invariant metrics on pseudoconvex domains of dimension two. Math. Z. 200, 429–466 (1989)

    Article  MathSciNet  Google Scholar 

  20. Gaussier, H.: Characterization of convex domains with noncompact automorphism group. Mich. Math. J. 44, 375–388 (1997)

    Article  MathSciNet  Google Scholar 

  21. Gaussier, H., Seshadri, H.: On the Gromov hyperbolicity of convex domains in \(\mathbb{C}^n\). Comput. Methods Funct. Theory 18, 617–641 (2018)

    Article  MathSciNet  Google Scholar 

  22. Ghys, É., De La Harpe, P.: (éditeurs) Sur les groupes hyperboliques, d’aprés Mikhael Gromov, avec la collaboration de W. Ballman, A. Haefliger, E. Salem, R. Strebel et M. Troyanov, Progress in Mathematics, Volume 83, Birkhaüser (1990)

  23. Survey, A., Isaev, A.V., Krantz, S.G.: Domains with non-compact automorphism group. Adv. Math. 146, 1–38 (1999)

    Article  MathSciNet  Google Scholar 

  24. Karlsson, A.: Non-expanding maps and Busemann functions. Ergodic Theory Dyn. Syst. 21, 1447–1457 (2001)

    Article  MathSciNet  Google Scholar 

  25. Kobayashi, S.: Hyperbolic Manifolds and Holomorphic Mappings: An Introduction (second edition) (2005)

  26. Kohn, J.J.: Boundary behavior of \(\delta \) on weakly pseudo-convex manifolds of dimension two. J. Differ. Geom. 6(4), 523–542 (1972). https://doi.org/10.4310/jdg/1214430641

    Article  MathSciNet  MATH  Google Scholar 

  27. Nikolov, N., Thomas, P.J., Trybula, M.: Gromov (non)hyperbolicity of certain domains in \(\mathbb{C}^2\). Forum Math. 28, 4 (2016)

    Article  Google Scholar 

  28. Van Thu, N., Duc, M.A.: On the automorphism groups of models in \({\mathbb{C}}^2\). Acta Math. Vietnam 41, 457–470 (2016)

    Article  MathSciNet  Google Scholar 

  29. Venturini, S.: Pseudodistances and pseudometrics on real and complex manifolds. Ann. Mat. Pura Appl 154(4), 385–402 (1989)

    Article  MathSciNet  Google Scholar 

  30. Yu, J.: Weighted boundary limits of the generalized Kobayashi–Royden metrics on weakly pseudoconvex domains. Trans. Am. Math. Soc. 347, 587–614 (1995)

    Article  MathSciNet  Google Scholar 

  31. Zimmer, A.: Gromov hyperbolicity and the Kobayashi metric on convex domains of finite type. Math. Ann. (2014)

  32. Zimmer, A.: Gromov hyperbolicity, the Kobayashi metric, and \( {\mathbb{C}}\)-convex sets. Trans. Am. Math. Soc. 369, 20 (2017)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The author would like to express his gratitude to Prof. Hervé Gaussier for his availability and for introducing him to Gromov hyperbolicity theory, and to Prof. Filippo Bracci for useful discussions and suggestions.

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  1. Dipartimento di Matematica, Università di Roma “Tor Vergata”, Via Della Ricerca Scientifica 1, 00133, Rome, Italy

    Matteo Fiacchi

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  1. Matteo Fiacchi
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Correspondence to Matteo Fiacchi.

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Communicated by Ngaiming Mok.

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Partially supported by PRIN Real and Complex Manifolds: Topology, Geometry and holomorphic dynamics n. 2017JZ2SW5 and by MIUR Excellence Department Project awarded to the Department of Mathematics, University of Rome Tor Vergata, CUP E83C18000100006.

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Fiacchi, M. Gromov hyperbolicity of pseudoconvex finite type domains in \({\mathbb {C}}^2\). Math. Ann. 382, 37–68 (2022). https://doi.org/10.1007/s00208-020-02135-w

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