Advertisement

Journal d’Analyse Mathématique

, Volume 100, Issue 1, pp 375–396 | Cite as

On maps almost quasi-conformally close to quasi-isometries

  • Vladimir Mikhaelovich Miklyukov
Article

Abstract

We prove thatW loc 1,n -maps almost quasi-conformally close to quasi-isometries are quasi-isometric under appropriate assumptions. Estimates of the inner distances and applications to the implicit function theorem are given.

Keywords

Jacobi Matrix Lipschitz Condition Implicit Function Theorem Quasiregular Mapping Dimensional Ball 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    F.G. Avkhadiev,Conformal Mappings and Boundary Problems, Kazanskii fond “Matematika,” Kazan’, 1996.zbMATHGoogle Scholar
  2. [2]
    E.D. Callender,Hölder-continuity of n-dimensional quasi-conformal mappings, Pacific J. Math.10 (1960), 499–515.MathSciNetzbMATHGoogle Scholar
  3. [3]
    H. Cartan,Calcul differentiel. Formes différentielles, Collection méthodes, Hermann, Paris, 1967.Google Scholar
  4. [4]
    F.H. Clarke,On the inverse function theorem, Pacific J. Math.64 (1976), 97–102.MathSciNetzbMATHGoogle Scholar
  5. [5]
    M. Cristea,Local inversion theorems and implicit function theorems without assuming continuous differentiability, Bull. Math. Soc. Sci. Math Roumanie (N.S.)36(84) (1992), 227–236.MathSciNetzbMATHGoogle Scholar
  6. [6]
    M. Cristea,Teoria topologică a functiilor analitice, Edit. Univer. din Bucure§ti, 1999.Google Scholar
  7. [7]
    H. Federer,Geometric Measure Theory, Springer-Verlag, Berlin, 1969.zbMATHGoogle Scholar
  8. [8]
    M. Gromov,Metric Structures for Riemannian and Non-Riemannian Spaces, Birkhäuser Bostom, Boston, MA, 1999.zbMATHGoogle Scholar
  9. [9]
    J. Heinonen,Lectures on Lipschitz Analysis, University of Jyväskylä, Jyväskylä, 2005.zbMATHGoogle Scholar
  10. [10]
    J. Heinonen and T. Kilpeläinen,BLD-mappings in W 2,2 are locally invertible, Math. Ann.318 (2000), 391–396.CrossRefMathSciNetzbMATHGoogle Scholar
  11. [11]
    J. Heinonen, T. Kilpeläinen and O. Martio,Nonlinear Potential Theory of Degenerate Elliptic Equation, Oxford University Press, Oxford, 1993.Google Scholar
  12. [12]
    R.A. Horn and C.R. Johnson,Matrix Analysis, Cambridge University Press, Cambridge, 1986.Google Scholar
  13. [13]
    F. John,On quasi-isometric mappings, I, Comm. Pure Appl. Math.21 (1968), 77–110.CrossRefMathSciNetzbMATHGoogle Scholar
  14. [14]
    M.A. Lavrentiev,On a differential indication of homeomorphic maps of three dimensional domains, Dokl. Akad. Nauk SSSR20 (1938), 241–242.Google Scholar
  15. [15]
    A. V. Lygin and V.M. Miklyukov,Distortion triangles under quasi-isometries, Advances in Grid Generation, ed. Olga V. Ushakova, 2005, pp. 57–72.Google Scholar
  16. [16]
    O. Martio, V.M. Miklyukov and M. Vuorinen,Morrey’s Lemma on Riemannian manifolds, Rev. Roumaine Math. Pures Appl.43 (1998), 183–209.MathSciNetzbMATHGoogle Scholar
  17. [17]
    O. Martio, S. Rickman and J. Väisälä,Topological and metric properties of quasiregular mappings, Ann. Acad. Sci. Fenn. Ser. A I Math.488 (1971).Google Scholar
  18. [18]
    V.M. Miklyukov,Conformal Mappings of Nonregular Surfaces and its Applications, Volgograd State University Press, Volgograd, 2005.Google Scholar
  19. [19]
    F. Nevanlinna,Über die Umkehrung differenzierbarer Abbildungen, Ann. Acad. Sci. Fenn. Ser. A. I.245 (1957).Google Scholar
  20. [20]
    T. Parthasarathy,On Global Univalence Theorems, Lectures Notes in Mathematics977, Springer-Verlag, Berlin-Heidelberg-New York, 1983.zbMATHGoogle Scholar
  21. [21]
    B.H. Pourcian,Analysis and optimization of Lipschitz continuous mappings, J. Optim. Theory Appl.22 (1977), 311–351.CrossRefGoogle Scholar
  22. [22]
    B. Pourciau,Global invertibility of nonsmooth mappings, J. Math. Anal. Appl.131 (1988), 170–179.CrossRefMathSciNetzbMATHGoogle Scholar
  23. [23]
    Yu.G. Reshetnyak,Space Mappings with Bounded Distortion, American Mathematical Society, Providence, RI, 1989.zbMATHGoogle Scholar
  24. [24]
    J. Warga,Fat homeomorphisms and unbounded derivate containers, J. Math. Anal. Appl.81 (1981), 545–560.CrossRefMathSciNetzbMATHGoogle Scholar
  25. [25]
    I.V. Zhuravlev and A.Yu. Igumnov,On implicit functions, Proceedings of Department of Mathematical Analysis and Function Theory, volgograd State University Press, Volgograd (2002), 41–46.Google Scholar
  26. [26]
    I.V. Zhuravlev, A.Yu. Igumnov and V.M. Miklyukov,An implicit function theorem, Rocky Mountain J. Math.36 (2006), 357–365.CrossRefMathSciNetzbMATHGoogle Scholar
  27. [27]
    V.A. Zorich,Lavrentiev theorem on quasiconformal maps of space, Math. Sb.74 (1967), 417–432.Google Scholar

Copyright information

© Hebrew University 2006

Authors and Affiliations

  1. 1.Volgograd State UniversityUniversitetskii Prospect 100VolgogradRussia

Personalised recommendations