On the isotopy problem for quasiconformal mappings

  • V. A. Zorich


The question of the isotopy of a quasiconformal mapping and its special aspects in dimension greater than 2 are considered. It is shown that an arbitrary quasiconformal mapping of a ball has an isotopy to the identity map such that the coefficient of quasiconformality (dilatation) of the mapping varies continuously and monotonically. In contrast to the planar case, in dimension higher than 2 such an isotopy is not possible in an arbitrary domain. Examples showing specific features of the multidimensional case are given. In particular, they show that even when such an isotopy exists, it is not always possible to perform an isotopy so that the coefficient of quasiconformality approaches 1 monotonically at each point in the source domain.


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, Princeton, NJ, 1966).MATHGoogle Scholar
  2. 2.
    L. Ahlfors and L. Bers, “Riemann’s mapping theorem for variable metrics,” Ann. Math., Ser. 2, 72 (2), 385–404 (1960).MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. 2: Partial Differential Equations (Interscience, New York, 1962).MATHGoogle Scholar
  4. 4.
    S. K. Donaldson and D. P. Sullivan, “Quasiconformal 4-manifolds,” Acta Math. 163 (3–4), 181–252 (1989).MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    M. H. Freedman and Z.-X. He, “Divergence-free fields: Energy and asymptotic crossing number,” Ann. Math., Ser. 2, 134 (1), 189–229 (1991).MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    M. H. Freedman and Z.-X. He, “Links of tori and the energy of incompressible flows,” Topology 30 (2), 283–287 (1991).MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    G. J. Martin, “The Teichmüller problem for mean distortion,” Ann. Acad. Sci. Fenn., Math. 34, 233–247 (2009).MathSciNetMATHGoogle Scholar
  8. 8.
    J. Milnor, Dynamics in One Complex Variable: Introductory Lectures (Vieweg, Wiesbaden, 1999).MATHGoogle Scholar
  9. 9.
    I. N. Vekua, Generalized Analytic Functions (Fizmatgiz, Moscow, 1959; Pergamon Press, Oxford, 1962).MATHGoogle Scholar
  10. 10.
    V. A. Zorich, “Quasi-conformal maps and the asymptotic geometry of manifolds,” Usp. Mat. Nauk 57 (3), 3–28 (2002) [Russ. Math. Surv. 57, 437–462 (2002)].CrossRefGoogle Scholar
  11. 11.
    V. A. Zorich, “Some observations concerning multidimensional quasiconformal mappings,” Mat. Sb. 208 (3), 72–95 (2017) [Sb. Math. 208, 377–398 (2017)].MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2017

Authors and Affiliations

  1. 1.Faculty of Mechanics and MathematicsMoscow State UniversityMoscowRussia

Personalised recommendations