Abstract
Given a nonempty compact set \( E \) in a proper subdomain \( \Omega \) of the complex plane, we denote the diameter of \( E \) and the distance from \( E \) to the boundary of \( \Omega \) by \( d(E) \) and \( d(E,\partial\Omega) \), respectively. The quantity \( d(E)/d(E,\partial\Omega) \) is invariant under similarities and plays an important role in geometric function theory. In case \( \Omega \) has the hyperbolic distance \( h_{\Omega}(z,w) \), we consider the infimum \( \kappa(\Omega) \) of the quantity \( h_{\Omega}(E)/\log(1+d(E)/d(E,\partial\Omega)) \) over compact subsets \( E \) of \( \Omega \) with at least two points, where \( h_{\Omega}(E) \) stands for the hyperbolic diameter of \( E \). Let the upper half-plane be \( \). We show that \( \kappa(\Omega) \) is positive if and only if the boundary of \( \Omega \) is uniformly perfect and \( \kappa(\Omega)\leq\kappa() \) for all \( \Omega \), with equality holding precisely when \( \Omega \) is convex.
Similar content being viewed by others
References
Beardon A. F. and Minda D., “The hyperbolic metric and geometric function theory,” in: Proc. Int. Workshop Quasiconformal Mappings and their Applications (IWQCMA05) (S. Ponnusamy, T. Sugawa and Vuorinen M., eds.) (2006), 9–56.
Keen L. and Lakic N., Hyperbolic Geometry from a Local Viewpoint, Cambridge Univ., Cambridge (2007) (Lond. Math. Soc. Stud. Texts; Vol. 68).
Beardon A. F. and Pommerenke Ch., “The Poincaré metric of plane domains,” J. Lond. Math. Soc. (2), vol. 18, 475–483 (1978).
Gehring F. W. and Hag K., The Ubiquitous Quasidisk, Amer. Math. Soc., Providence (2012) (Math. Surv. Monogr.; Vol. 184).
Gehring F. W., Palka B. P., “Quasiconformally homogeneous domains,” J. Anal. Math., vol. 30, 172–199 (1976).
Sugawa T. and Vuorinen M., “Some inequalities for the Poincaré metric of plane domains,” Math. Z., vol. 250, no. 4, 885–906 (2005).
Harmelin R. and Minda D., “Quasi-invariant domain constants,” Israel J. Math., vol. 77, 115–127 (1992).
Mejía D. and Minda D., “Hyperbolic geometry in \( k \)-convex regions,” Pacific J. Math., vol. 141, 333–354 (1990).
Avkhadiev F. G. and Wirths K.-J., Schwarz–Pick Type Inequalities. Frontiers in Mathematics, Birkhäuser, Basel (2009).
Garnett J. B. and Marshall D. E., Harmonic Measure, Cambridge Univ., Cambridge (2005).
Pommerenke Ch., “Uniformly perfect sets and the Poincaré metric,” Arch. Math., vol. 32, 192–199 (1979).
Pommerenke Ch., “On uniformly perfect sets and Fuchsian groups,” Analysis, vol. 4, 299–321 (1984).
Sugawa T., “Various domain constants related to uniform perfectness,” Complex Variables Theory Appl., vol. 36, 311–345 (1998).
Sugawa T., “Uniformly perfect sets: analytic and geometric aspects (Japanese),” Sugaku Expo., vol. 16, 225–242 (2003).
Bridgeman M. and Canary R. D., “Uniformly perfect domains and convex hulls: improved bounds in a generalization of a theorem of Sullivan,” Pure Appl. Math. Q., vol. 9, no. 1, 49–71 (2013).
Stankewitz R., Sugawa T., and Sumi H., “Hereditarily non uniformly perfect sets,” Discrete Contin. Dyn. Syst., vol. 12, no. 8, 2391–2402 (2019).
Wang X. and Zhou Q., “Quasimöbius maps, weakly quasimöbius maps and uniform perfectness in quasi-metric spaces,” Ann. Acad. Sci. Fenn. Math., vol. 42, no. 1, 257–284 (2017).
Hariri P., Klén R., and Vuorinen M., Conformally Invariant Metrics and Quasiconformal Mappings. Springer Monogr. Math., Springer, Berlin (2020).
Gehring F. W. and Osgood B. G., “Uniform domains and the quasi-hyperbolic metric,” J. Anal. Math., vol. 36, 50–74 (1979).
Golberg A., Sugawa T., and Vuorinen M., “Teichmüller’s theorem in higher dimensions and its applications,” Comput. Methods Funct. Theory, vol. 20, no. 3, 539–558 (2020).
Keogh F. R., “A characterization of convex domains in the plane,” Bull. Lond. Math. Soc., vol. 8, 183–185 (1976).
Beardon A. F., The Geometry of Discrete Groups. Grad. Texts Math.; Vol. 91, Springer, New York (1983).
Acknowledgments
The authors would like to thank the referee for detailed and constructive corrections.
Funding
The authors were supported in part by the JSPS KAKENHI (Grant JP17H02847).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Sibirskii Matematicheskii Zhurnal, 2021, Vol. 62, No. 4, pp. 845–863. https://doi.org/10.33048/smzh.2021.62.412
Rights and permissions
About this article
Cite this article
Rainio, O., Sugawa, T. & Vuorinen, M. Intrinsic Geometry and Boundary Structure of Plane Domains. Sib Math J 62, 691–706 (2021). https://doi.org/10.1134/S0037446621040121
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0037446621040121