Abstract
In this paper, we prove that the identity map for the smoothly bounded pseudoconvex domain of finite type in \({\mathbb {C}}^2\) extends to a bi-Hölder map between the Euclidean boundary and Gromov boundary. As an application, we show the bi-Hölder boundary extensions for quasi-isometries between these domains. Moreover, we get a more accurate index of the Gehring–Hayman type theorem for the bounded m-convex domains with Dini-smooth boundary.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Balogh, Z.M., Bonk, M.: Gromov hyperbolicity and the Kobayashi metric on strictly pseudoconvex domains. Comment. Math. Helv. 75(3), 504–533 (2000)
Barth, T.J.: Convex domains and Kobayashi hyperbolicity. Proc. Am. Math. Soc. 79(4), 556–558 (1980)
Bharali, G., Zimmer, A.: Goldilocks domains, a weak notion of visibility, and applications. Adv. Math. 310(2), 377–425 (2017)
Bloom, T., Graham, I.: A geometric characterization of points of type \(m\) on real submanifolds of \({\mathbb{C} }^n\). J. Differ. Geom. 12(2), 171–182 (1977)
Bonk, M., Schramm, O.: Embeddings of Gromov hyperbolic spaces. Geom. Funct. Anal. 10, 266–306 (2000)
Bracci, F., Nikolov, N., Thomas, P.J.: Visibility of Kobayashi geodesics in convex domains and related properties. Math. Z. 301(2), 2011–2035 (2022)
Bridson, M.R., Haefliger, A.: Metric spaces of non-positive curvature. In: Fundamental Principles of Mathematical Sciences, vol. 319. Springer, Berlin (1999)
Catlin, D.W.: Estimates of invariant metrics on pseudoconvex domains of dimension two. Math. Z. 200(3), 429–466 (1989)
Cho, S.: A lower bound on the Kobayashi metric near a point of finite type in \({\mathbb{C} }^n\). J. Geom. Anal. 2(4), 317–325 (1992)
Fiacchi, M.: Gromov hyperbolicity of pseudoconvex finite type domains in \({\mathbb{C} }^2\). Math. Ann. 382(1), 37–68 (2022)
Forstnerič, F.: Proper holomorphic mappings: a survey. Several Complex Variables (Stockholm, 1987/1988). Mathematical Notes, vol. 38. Princeton University Press, Princeton, pp. 297–363 (1993)
Forstnerič, F., Rosay, J.P.: Localization of the Kobayashi metric and the boundary continuity of proper holomorphic mappings. Math. Ann. 279(2), 239–252 (1987)
Graham, I.: Sharp constants for the Koebe theorem and for estimates of intrinsic metrics on convex domains. Several Complex Variables and Complex Geometry, Part 2 (Santa Cruz, CA, 1989). Proceedings of Symposia in Pure Mathematics, vol. 52, pp. 233–238 (1991)
Herbort, G.: Estimation on invariant distances on pseudoconvex domains of finite type in dimension two. Math. Z. 251(3), 673–703 (2005)
Liu, J., Wang, H., Zhou, Q.: Bi-Hölder extensions of quasi-isometries on complex domains. J. Geom. Anal. 32(2), 38 (2022)
Mercer, P.R.: Complex geodesics and iterates of holomorphic maps on convex domains in \({\mathbb{C} }^n\). Trans. Am. Math. Soc. 338(1), 201–211 (1993)
Nikolov, N., Andreev, L.: Estimates of the Kobayashi and quasi-hyperbolic distances. Ann. Mat. Pura Appl. 196(1), 43–50 (2017)
Nikolov, N., Ökten, A.Y.: Strongly Goldilocks domains, quantitative visibility, and applications. Preprint (2022). arXiv:2206.08344
Nikolov, N., Thomas, P. J.: Comparison of the real and the complex Green functions, and sharp estimates of the Kobayashi distance. Ann. Scuola Norm. Superior. Pisa Classe Sci. 18(3), 1125–1143 (2018)
Nikolov, N., Thomas, P.J.: Quantitative localization and comparison of invariant distances of domains in \({\mathbb{C} }^n\). J. Geom. Anal. 33(1), 35 (2023)
Wang, H.: Estimates of the Kobayashi metric and Gromov hyperbolicity on convex domains of finite type. Preprint (2022). arXiv:2211.10662
Zimmer, A.: Gromov hyperbolicity and the Kobayashi metric on convex domains of finite type. Math. Ann. 365(3–4), 1425–1498 (2016)
Zimmer, A.: Subelliptic estimates from Gromov hyperbolicity. Adv. Math. 402(94), 108334 (2022)
Acknowledgements
The authors would like to thank the referee for a careful reading and valuable comments.
Funding
J. Liu is supported by National Key R &D Program of China (Grant No. 2021YFA1003100), NSFC (Grants Nos. 11925107 and 12226334), Key Research Program of Frontier Sciences, CAS (Grant No. ZDBS-LY-7002). H. Wang is supported by the Fundamental Research Funds for the Central Universities (Grant No. 500422379).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Liu, J., Pu, X. & Wang, H. Bi-Hölder Extensions of Quasi-isometries on Pseudoconvex Domains of Finite Type in \({\mathbb {C}}^2\). J Geom Anal 33, 152 (2023). https://doi.org/10.1007/s12220-023-01204-1
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s12220-023-01204-1