Abstract
In this chapter we shall consider some geometric issues related to the hyperbolic or quasihyperbolic metric. We begin with several comparison results for the quasihyperbolic metric. Here an important fact is that various metrics may be comparable in some but not in all domains.
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References
Ahlfors, L. V.Möbius transformations in several dimensions. Ordway Professorship Lectures in Mathematics. University of Minnesota, School of Mathematics, Minneapolis, Minn., 1981.
Ahlfors, L. V.Collected papers. Vol. 1,2. Contemporary Mathematicians. Birkhäuser, Boston, Mass., 1982. 1929–1955, Edited with the assistance of Rae Michael Shortt.
Anderson, G. D., Vamanamurthy, M. K., and Vuorinen, M. K.Conformal invariants, inequalities, and quasiconformal maps. Canadian Mathematical Society Series of Monographs and Advanced Texts. John Wiley & Sons, Inc., New York, 1997. With 1 IBM-PC floppy disk (3.5 inch; HD), A Wiley-Interscience Publication.
Armitage, D. H., and Gardiner, S. J.Classical potential theory. Springer Monographs in Mathematics. Springer-Verlag London, Ltd., London, 2001.
Beardon, A. F.The geometry of discrete groups, vol. 91 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1983.
Berger, M.Geometry. I, II. Universitext. Springer-Verlag, Berlin, 1987. Translated from the French by M. Cole and S. Levy.
Coxeter, H. S. M., and Greitzer, S. L.Geometry revisited, vol. 19 of New Mathematical Library. Random House, Inc., New York, 1967.
de Guzmán, M.Differentiation of integrals in \(\mathbb {R}^{n}\). Lecture Notes in Mathematics, Vol. 481. Springer-Verlag, Berlin-New York, 1975. With appendices by Antonio Córdoba, and Robert Fefferman, and two by Roberto Moriyón.
Garnett, J. B., and Marshall, D. E.Harmonic measure, vol. 2 of New Mathematical Monographs. Cambridge University Press, Cambridge, 2008. Reprint of the 2005 original.
Gehring, F. W., and Hag, K.The ubiquitous quasidisk, vol. 184 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2012. With contributions by Ole Jacob Broch.
Gehring, F. W., and Osgood, B. G. Uniform domains and the quasihyperbolic metric. J. Analyse Math. 36 (1979), 50–74 (1980).
Gilbarg, D., and Trudinger, N. S.Elliptic partial differential equations of Second order. Springer-Verlag, Berlin-New York, 1977. Grundlehren der Mathematischen Wissenschaften, Vol. 224.
Gol′dshteı̆n, V. M.,andReshetnyak, Y. G.Quasiconformal mappings and Sobolev spaces, vol. 54 of Mathematics and its Applications (Soviet Series). Kluwer Academic Publishers Group, Dordrecht, 1990. Translated and revised from the 1983 Russian original, Translated by O. Korneeva.
Harmaala, E., and Klén, R. Ptolemy constant and uniformity. arXiv:1604.0536.
Hästö, P. A., Klén, R., Sahoo, S. K., and Vuorinen, M. K. Geometric properties of φ-uniform domains. J. Anal. 24, 1 (2016), 57–66.
Hayman, W. K.Subharmonic functions. Vol. 2, vol. 20 of London Mathematical Society Monographs. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London, 1989.
Herron, D. A., and Minda, D. Comparing invariant distances and conformal metrics on Riemann surfaces. Israel J. Math. 122 (2001), 207–220.
Herron, D. A., and Vuorinen, M. K. Positive harmonic functions in uniform and admissible domains. Analysis 8, 1–2 (1988), 187–206.
Huang, M., Wang, X., Ponnusamy, S., and Sahoo, S. K. Uniform domains, John domains and quasi-isotropic domains. J. Math. Anal. Appl. 343, 1 (2008), 110–126.
John, F. Rotation and strain. Comm. Pure Appl. Math. 14 (1961), 391–413.
Jones, P. W. Extension theorems for BMO. Indiana Univ. Math. J. 29, 1 (1980), 41–66.
Jones, P. W. Quasiconformal mappings and extendability of functions in Sobolev spaces. Acta Math. 147, 1–2 (1981), 71–88.
Käenmäki, A., Lehrbäck, J., and Vuorinen, M. K. Dimensions, Whitney covers, and tubular neighborhoods. Indiana Univ. Math. J. 62, 6 (2013), 1861–1889.
Klén, R. On hyperbolic type metrics. Ann. Acad. Sci. Fenn. Math. Diss., 152 (2009), 49. Dissertation, University of Turku, Turku, 2009.
Klén, R., Li, Y., Sahoo, S. K., and Vuorinen, M. K. On the stability of φ-uniform domains. Monatsh. Math. 174, 2 (2014), 231–258.
Landkof, N. S.Foundations of modern potential theory. Springer-Verlag, New York-Heidelberg, 1972. Translated from the Russian by A. P. Doohovskoy, Die Grundlehren der mathematischen Wissenschaften, Band 180.
Lindén, H. Quasihyperbolic geodesics and uniformity in elementary domains. Ann. Acad. Sci. Fenn. Math. Diss., 146 (2005), 50. Dissertation, University of Helsinki, Helsinki, 2005.
Martio, O., and Sarvas, J. Injectivity theorems in plane and space. Ann. Acad. Sci. Fenn. Ser. A I Math. 4, 2 (1979), 383–401.
Martio, O., and Vuorinen, M. K. Whitney cubes, p-capacity, and Minkowski content. Exposition. Math. 5, 1 (1987), 17–40.
Mattila, P.Geometry of sets and measures in Euclidean spaces, Fractals and rectifiability, vol. 44 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1995.
Näkki, R. Extension of Loewner’s capacity theorem. Trans. Amer. Math. Soc. 180 (1973), 229–236.
Näkki, R., and Väisälä, J. John disks. Exposition. Math. 9, 1 (1991), 3–43.
Ransford, T.Potential theory in the complex plane, vol. 28 of London Mathematical Society Student Texts. Cambridge University Press, Cambridge, 1995.
Reshetnyak, Y. G. Stability in Liouville’s theorem on conformal mappings of a space for domains with a nonsmooth boundary. Sibirsk. Mat. Ž. 17, 2 (1976), 361–369, 479.
Reshetnyak, Y. G.Space mappings with bounded distortion, vol. 73 of Translations of Mathematical Monographs. American Mathematical Society, Providence, RI, 1989. Translated from the Russian by H. H. McFaden.
Reshetnyak, Y. G.Stability theorems in geometry and analysis, vol. 304 of Mathematics and its Applications. Kluwer Academic Publishers Group, Dordrecht, 1994. Translated from the 1982 Russian original by N. S. Dairbekov and V. N. Dyatlov, and revised by the author, Translation edited and with a foreword by S. S. Kutateladze.
Seittenranta, P. Möbius-invariant metrics. Math. Proc. Cambridge Philos. Soc. 125, 3 (1999), 511–533.
Stein, E. M.Singular integrals and differentiability properties of functions. Princeton Mathematical Series, No. 30. Princeton University Press, Princeton, N.J., 1970.
Trotsenko, D. A. Properties of regions with a nonsmooth boundary. Sibirsk. Mat. Zh. 22, 4 (1981), 221–224, 232.
Väisälä, J. Uniform domains. Tohoku Math. J. (2) 40, 1 (1988), 101–118.
Väisälä, J. Free quasiconformality in Banach spaces. II. Ann. Acad. Sci. Fenn. Ser. A I Math. 16, 2 (1991), 255–310.
Väisälä, J. Domains and maps. In Quasiconformal space mappings, vol. 1508 of Lecture Notes in Math. Springer, Berlin, 1992, pp. 119–131.
Vuorinen, M. K. Capacity densities and angular limits of quasiregular mappings. Trans. Amer. Math. Soc. 263, 2 (1981), 343–354.
Vuorinen, M. K. On the Harnack constant and the boundary behavior of Harnack functions. Ann. Acad. Sci. Fenn. Ser. A I Math. 7, 2 (1982), 259–277.
Vuorinen, M. K. Conformal invariants and quasiregular mappings. J. Analyse Math. 45 (1985), 69–115.
Zhang, X., Klén, R., Suomala, V., and Vuorinen, M. K. Volume growth of quasihyperbolic balls. Mat. Sb. 208, 6 (2017), 170–182.
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Hariri, P., Klén, R., Vuorinen, M. (2020). Metrics and Geometry. In: Conformally Invariant Metrics and Quasiconformal Mappings. Springer Monographs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-030-32068-3_6
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