Abstract
We consider two conformally invariant metrics in proper subdomains of euclideann-spaceR n. We show that Lipschitz mappings in these metrics include the class of quasiconformational mappings as a proper subclass, yet these Lipschitz mappings have many properties similar to those of quasiconformal mappings.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
[AVV1] G. D. Anderson, M. K. Vamanamurthy and M. Vuorinen,Dimension-free quasiconformal distorition in n-space, Trans. Am. Math. Soc.297 (1986), 687–706.
[AVV2] G. D. Anderson, M. K. Vamanamurthy and M. Vuorinen,Special functions of quasiconformal theory, Exposition. Math.7 (1989), 97–138.
[F1] J. Lelong-Ferrand,Invariants conformes globaux sur les varietés riemanniennes, J. Differ. Geom.8 (1973), 487–510.
[F2] J. Lelong-Ferrand,Construction de métriques pour lesquelles les transformations quasiconformes sont lipschitziennes, Symposia Mathematica Vol. XVIII, INDAM, Academic Press, London, 1976, pp. 407–420.
[F3] J. Ferrand,A characterization of quasiconformal mappings by the behaviour of a function of three points, Proc. 13th Rolf Nevanlinna Colloquium Joensuu (1987), Lecture Notes in Math. Vol. 1351, Springer-Verlag, Berlin, 1988, pp. 110–123.
[G1] F. W. Gehring,Symmetrization of rings in space, Trans. Am. Math. Soc.101 (1961), 499–519.
[G2] F. W. Gehring,Rings and quasiconformal mappings in space, Trans. Am. Math. Soc.103 (1962), 353–393.
[53] F. W. Gehring,Quasiconformal mappings in space, Bull. Am. Math. Soc.69 (1963), 146–164.
[GO] F. W. Gehring and B. G. Osgood,Uniform domains and the, quasihyperbolic metric, J. Analyse Math.36 (1979), 50–74.
[GP] F. W. Gehring and B. P. Palka,Quasiconformally homogeneous domains, J. Analyse Math.30 (1976), 172–199.
[GV] F. W. Gehring and J. Väisälä,The coefficient of quasiconformality of domains in space, Acta Math.114 (1965), 1–70.
[GOR] V. M. Gol’dstein and Yu. G. Reshetnyak,Introduction to the Theory of Functions with Generalized Derivatives and Quasiconformal Mappings (Russian), Izdat. “Nauka”, Moscow, 1983 (English translation in preparation).
[GOV] V. M. Gol’dstein and S. K. Vodop’yanov,Metric completion of a domain by using a conformal capacity invariant under quasiconformal mappings, Soviet Math. Dokl.19 (1978), No. 1, 158–161.
[K] J. Kelingos,Characterizations of quasiconformal mappings in terms of harmonic and hyperbolic measure, Ann. Acad. Sci. Fenn Ser. A I368 (1965), 3–16.
[LV] O. Lehto and K. I. Virtanen,Quasiconformal mappings in the plane, Grundlehr. Math. Wiss., Vol. 126, second ed., Springer-Verlag, Berlin, 1973.
[MAO] G. Martin, B. G. Osgood,The quasihyperbolic metric and the associated estimates on the hyperbolic metric, J. Analyse Math.47 (1986), 37–53.
[T] P. Tukia,On two dimensional quasiconformal groups, Ann. Acad. Sci. Fenn. Ser. A I5 (1980), 73–78.
[V1] J. Väisälä,Two new characterizations for quasiconformality, Ann. Acad. Sci. Fenn. Ser. A I362 (1965), 3–12.
[V2] J. Väisälä,Lectures on n-dimensional quasiconformal mappings, Lecture Notes in Math., Vol. 229, Springer-Verlag, Berlin, 1971.
[VU1] M. Vuorinen,Conformal invariants and quasiregular mappings, J. Analyse Math.45 (1985), 69–115.
[VU2] M. Vuorinen,On Teichmüller’s modulus problem in R n, Math Scand.63 (1988), 315–333.
[VU3] M. Vuorinen,Conformal geometry and quasiregular mappings, Lecture Notes in Math., Vol. 1319, Springer-Verlag, Berlin, 1988.
Author information
Authors and Affiliations
Additional information
Research supported in part by the U.S. National Science Foundation and the A. P. Sloan Foundation.
Research supported in part by the Alexander von Humboldt Foundation.
Rights and permissions
About this article
Cite this article
Ferrand, J., Martin, G.J. & Vuorinen, M. Lipschitz conditions in conformally invariant metrics. J. Anal. Math. 56, 187–210 (1991). https://doi.org/10.1007/BF02820464
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02820464