Journal d'Analyse Mathématique

, Volume 113, Issue 1, pp 265–291 | Cite as

Modulus of curve families and extremality of spiral-stretch maps

  • Zoltán M. Balogh
  • Katrin Fässler
  • Ioannis D. Platis


We develop a method using the modulus of curve families to study minimisation problems for the mean distortion functional in the class of finite distortion homeomorphisms. We apply our method to prove extremality of the spiral-stretch mappings defined on annuli in the complex plane. This generalises results of Gutlyanskiĭ and Martio [12] and Strebel [23].


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    L. V. Ahlfors, Lectures on Quasiconformal Mappings, D. Van Nostrand Co., 1966.Google Scholar
  2. [2]
    K. Astala, Area distortion of quasiconformal mappings, Acta Math. 173 (1994), 37–60.CrossRefMATHMathSciNetGoogle Scholar
  3. [3]
    K. Astala, T. Iwaniec and G. Martin, Deformations of Annuli with Smallest Mean Distortion, Arch. Ration. Mech. Anal. 195 (2009), 899–921.CrossRefMathSciNetGoogle Scholar
  4. [4]
    K. Astala, T. Iwaniec and G. Martin, Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane, Princeton University Press, Princeton, NJ, 2009.MATHGoogle Scholar
  5. [5]
    K. Astala, T. Iwaniec, G. J. Martin and J. Onninen, Extremal mappings of finite distortion, Proc. London Math. Soc. (3) 91 (2005), 655–702.CrossRefMATHMathSciNetGoogle Scholar
  6. [6]
    C. J. Bishop, V. Ya Gutlyanskiĭ, O. Martio and M. Vuorinen, On conformal dilatation in space, Int. J. Math. Math. Sci. 2003 (2003), 1397–1420.CrossRefMATHGoogle Scholar
  7. [7]
    M. A. Brakalova, Sufficient and necessary conditions for Conformality at a point I. Geometric viewpoint, Complex Var. Elliptic Equations 54 (2010), 137–155.CrossRefMathSciNetGoogle Scholar
  8. [8]
    M. A. Brakalova, Sufficient and necessary conditions for conformality II. Analytic viewpoint, Ann. Acad. Sci. Fenn. 35 (2010), 235–254.CrossRefMATHMathSciNetGoogle Scholar
  9. [9]
    B. Fuglede, Extremal length and functional completion, Acta Math. 98 (1957), 171–219.CrossRefMATHMathSciNetGoogle Scholar
  10. [10]
    F. P. Gardiner and N. Lakic, Quasiconformal Teichmüller Theory, American Mathematical Society, Providence, RI, 2000.MATHGoogle Scholar
  11. [11]
    F. W. Gehring, The Lp-integrability of the partial derivatives of a quasiconformal mapping, Acta Math. 130 (1973), 265–277.CrossRefMATHMathSciNetGoogle Scholar
  12. [12]
    V. Gutlyanskiĭ and O. Martio, Rotation estimates and spirals, Conform. Geom. Dyn. 5 (2001), 6–20 (electronic).CrossRefMATHMathSciNetGoogle Scholar
  13. [13]
    T. Iwaniec, P. Koskela and J. Onninen, Mappings of finite distortion: monotonicity and continuity, Invent. Math. 144 (2001), 507–531.CrossRefMATHMathSciNetGoogle Scholar
  14. [14]
    F. John, Rotation and strain, Comm. Pure Appl. Math. 14 (1961), 391–413.CrossRefMATHMathSciNetGoogle Scholar
  15. [15]
    J. Kauhanen, P. Koskela and J. Malý, Mappings of finite distortion: condition N, Michigan Math. J. 49 (2001), 169–181.CrossRefMATHMathSciNetGoogle Scholar
  16. [16]
    J. Kauhanen, P. Koskela and J. Malý, Mappings of finite distortion: discreteness and openness, Arch. Ration. Mech. Anal. 160 (2001), 135–151.CrossRefMATHMathSciNetGoogle Scholar
  17. [17]
    O. Lehto and K. I. Virtanen, Quasiconformal Mappings in the Plane, second edn., Springer-Verlag, New York, 1973.MATHGoogle Scholar
  18. [18]
    G. J. Martin, The Teichműller problem for mean distortion, Ann. Acad. Sci. Fenn. Math. 34 (2009), 233–247.MATHMathSciNetGoogle Scholar
  19. [19]
    O. Martio, Nonlinear potential theory in metric spaces, in Topics in Mathematical Analysis, World Scientific, Hackensack, NJ, 2008, pp. 29–60.CrossRefGoogle Scholar
  20. [20]
    O. Martio, V. Ryazanov, U. Srebro and E. Yakubov, On Q-homeomorphisms, Ann. Acad. Sci. Fenn. Math. 30 (2005), 49–69.MATHMathSciNetGoogle Scholar
  21. [21]
    O. Martio, V. Ryazanov, U. Srebro and E. Yakubov, Moduli in Modern Mapping Theory, Springer, New York, 2009.MATHGoogle Scholar
  22. [22]
    J. C. C. Nitsche, On the module of doubly-connected regions under harmonic mappings, Amer. Math. Monthly 69 (1962), 781–782.CrossRefMATHMathSciNetGoogle Scholar
  23. [23]
    K. Strebel, Zur Frage der Eindeutigkeit extremaler quasikonformer Abbildungen des Einheitskreises, Comment. Math. Helv. 36 (1961/1962), 306–323.CrossRefMathSciNetGoogle Scholar
  24. [24]
    K. Strebel, Extremal quasiconformal mappings, Results Math. 10 (1986), 168–210.MATHMathSciNetGoogle Scholar
  25. [25]
    O. Teichmüller, Extremale quasikonforme Abbildungen und quadratische Differentiale, Abh. Preuss. Akad. Wiss. Math.-Nat. Kl. 1939 (1940), no. 22.Google Scholar
  26. [26]
    O. Teichmüller, Bestimmung der extremalen quasikonformen Abbildung bei geschlossenen orientierten Riemannschen Flächen, Abh. Preuss. Akad. Wiss. Math.-Nat. Kl. 1943 (1943), no. 4.Google Scholar
  27. [27]
    J. Väisälä, Lectures on n-Dimensional Quasiconformal Mappings, Lecture Notes in Mathematics 229, Springer-Verlag, Berlin, 1971.Google Scholar

Copyright information

© Hebrew University Magnes Press 2011

Authors and Affiliations

  • Zoltán M. Balogh
    • 1
  • Katrin Fässler
    • 1
  • Ioannis D. Platis
    • 2
  1. 1.Department of MathematicsUniversity of BernBernSwitzerland
  2. 2.Department of MathematicsUniversity of CreteHeraklion, CreteGreece

Personalised recommendations