Skip to main content
Log in

Stability of classes of higher-dimensional holomorphic maps. 1. The stability concept. Liouville's theorem

  • Published:
Siberian Mathematical Journal Aims and scope Submit manuscript

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Literature Cited

  1. M. A. Lavrent'ev, “Sur une classe de représentations continues,” Mat. Sb.,42, 407–423 (1935).

    Google Scholar 

  2. M. A. Lavrent'ev, “Quasiconformal maps,” in: Proceedings of the 3rd All-Union Math. Congress, Vol. 3, Survey Reports [in Russian], Izd. Akad. Nauk SSSR, Moscow (1958), pp. 198–208.

    Google Scholar 

  3. M. A. Lavrent'ev, “On stability in Liouville's theorem” Dokl. Akad. Nauk SSSR,95, No. 5, 925–926 (1954).

    Google Scholar 

  4. P. P. Belinskii, “On distortion under quasiconformal maps,” Dokl. Akad. Nauk SSSR,91, No. 5, 997–998 (1953).

    Google Scholar 

  5. P. P. Belinskii, General Properties of Quasiconformal Maps [in Russian], Nauka, Novosibirsk (1974).

    Google Scholar 

  6. P. P. Belinskii, “On the continuity of quasiconformal maps in space, and on Liouville's theorem,” Dokl. Akad. Nauk SSSR,147, No. 5, 1003–1004 (1962).

    Google Scholar 

  7. P. P. Belinskii, “Compactness in Liouville's theorem on quasiconformal maps of spaces of dimension n≥3,”, in: International Math. Congress, Theses of Short Reports, Sec. 4. Classical Analysis [in Russian], Moscow (1966), p. 35.

  8. P. P. Belinskii, “On stability in Liouville's theorem on quasiconformal maps in space,” in: Some Problems in Mathematics and Physics [in Russian], Leningrad (1970), pp. 88–102.

  9. P. P. Belinskii, “On the order of approximation of quasiconformal maps in space to conformal maps,” Dokl. Akad. Nauk SSSR,200, No. 4, 759–761 (1971).

    Google Scholar 

  10. P. P. Belinskii, “On the order of approximation of a quasiconformal map in space to a conformal one,” Sib. Mat. Zh.,14, No. 3, 475–483 (1973).

    Google Scholar 

  11. Yu. G. Reshetnyak, “Stability in the Liouville's theorem on conformal maps in space,” in: Some Problems in Mathematics and Physics [in Russian], Novosibirsk (1961), pp. 219–223.

  12. Yu. G. Reshetnyak, “Stability in the theorem of Liouville on conformal mappings of space,” Dokl. Akad. Nauk SSSR,152, No. 2, 286–287 (1963).

    Google Scholar 

  13. Yu. G. Reshetnyak, “Stability of conformal mappings in multidimensional spaces,” Sib. Mat. Zh.,8, No. 1, 91–114 (1967).

    Google Scholar 

  14. Yu. G. Reshetnyak, “Stability theorems for mappings with bounded distortion,” in: The 2nd All-Union Symposium on Geometry in the Large [in Russian], Petrozavodsk (1967), pp. 54–56.

  15. Yu. G. Reshetnyak, “Stability theorems for mappings with finite distortion,” Sib. Mat. Zh.,9, No. 3, 667–684 (1968).

    Google Scholar 

  16. Yu. G. Reshetnyak, “Stability estimates in Liouville's theorem on conformal maps in space,” in: The 3rd All-Union Symposium on Geometry in the Large [in Russian], Petrozavodsk (1969), p. 57.

  17. Yu. G. Reshetnyak, “On a stability estimate in Liouville's theorem on conformal maps of higher dimensional spaces,” Sib. Mat. Zh.,11, No. 5, 1121–1139 (1970).

    Google Scholar 

  18. Yu. G. Reshetnyak, “Stability in Liouville's theorem on conformal maps of domains with nonsmooth boundary in space,” Sib. Mat. Zh.,17, No. 2, 361–369 (1976).

    Google Scholar 

  19. Yu. G. Reshetnyak, “Stability estimates in Liouville's theorem and Lp-integrability of the derivatives of quasiconformal maps,” Sib. Mat. Zh.,17, No. 4, 868–896 (1976).

    Google Scholar 

  20. Yu. G. Reshetnyak, “Stability estimates in the class W 1p in Liouville's theorem on conformal maps of a closed domain,” Sib. Mat. Zh.,17, No. 6, 1382–1394 (1976).

    Google Scholar 

  21. Yu. G. Reshetnyak, “Stability theorems in certain problems of differential geometry an analysis,” Mat. Zametki,23, No. 5, 773–781 (1978).

    Google Scholar 

  22. I. N. Vekua, Generalized Analytic Functions, Pergamon (1962).

  23. L. V. Ahlfors, Lectures on Quasiconformal Maps, Van Nostrand, Princeton, New Jersey (1966).

    Google Scholar 

  24. F. Hausdorff, Set Theory, Chelsea Publ.

  25. Yu. G. Reshetnyak, “Liouville's theorm on conformal maps under minimal regularity assumptions,” Sib. Mat. Zh.,8, No. 4, 835–840 (1967).

    Google Scholar 

  26. S. L. Sobolev, Applications of Functional Analysis in Mathematical Physics, Amer. Math. Soc. (1969).

  27. L. Bers, F. John, and M. Shechter, Contributions to the Theory of Partial Differential Equations, Kraus Repr. (1954).

  28. B. V. Boyarskii, “Homeomorphic solutions of Beltrami systems,” Dokl. Akad. Nauk SSSR,102, No. 5, 661–664 (1955).

    Google Scholar 

  29. O. Lehto, “Remarks on the integrability of the derivatives of quasiconformal mappings,” Ann. Acad. Sci. Fenn. Ser. A, I,371, 1–8 (1965).

    Google Scholar 

  30. Yu. G. Reshetnyak, “Space maps with bounded distortion,” Sib. Mat. Zh.8, No. 3, No. 3, 629–658 (1967).

    Google Scholar 

  31. A. P. Kopylov, “On the approximation of quasiconformal maps in the space by smooth quasiconformal maps,” Sib. Mat. Zh.,13, No. 1, 94–106 (1972).

    Google Scholar 

  32. F. W. Gehring, “The L-integrability of the partial derivatives of a quasiconformal mapping,” Acta Math.,130, 265–277 (1973).

    Google Scholar 

  33. V. M. Gol'dshtein, “On the behavior of maps with bounded distortion when the distortion coefficient is close to one,” Sib. Mat. Zh.,12, No. 6, 1250–1258 (1971).

    Google Scholar 

  34. O. Martio, S. Rickman, and I. Vaisälä, “Topological and metric properties of quasiregular mappings,” Ann. Acad. Sci. Fenn. Ser. A, I,488, 1–31 (1971).

    Google Scholar 

  35. A. P. Kopylov, “On the behavior of quasiconformal maps on hyperplanes of space which are close to conformal maps,” Dokl. Akad. Nauk SSSR,209, No. 6, 1278–1280 (1973).

    Google Scholar 

Download references

Authors

Additional information

Institute of Mathematics, Siberian Branch, Academy of Sciences of the USSR, Novosibirsk. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 23, No. 2, pp. 83–111, March–April, 1982.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kopylov, A.P. Stability of classes of higher-dimensional holomorphic maps. 1. The stability concept. Liouville's theorem. Sib Math J 23, 203–224 (1982). https://doi.org/10.1007/BF00971693

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00971693

Keywords

Navigation