On quasiconformal maps with identity boundary values

Manojlović, V.; Vuorinen, M.

doi:10.1090/S0002-9947-2010-04994-3

On quasiconformal maps with identity boundary values
HTML articles powered by AMS MathViewer

by V. Manojlović and M. Vuorinen PDF

Trans. Amer. Math. Soc. 363 (2011), 2467-2479 Request permission

Abstract:

Quasiconformal homeomorphisms of the unit ball $B^n$ of ${\mathbb R}^n, n \ge 3,$ onto itself with identity boundary values are studied. A spatial analogue of Teichmüller’s theorem is proved.

References

G. D. Anderson and M. K. Vamanamurthy, An extremal displacement mapping in $n$-space, Complex analysis Joensuu 1978 (Proc. Colloq., Univ. Joensuu, Joensuu, 1978) Lecture Notes in Math., vol. 747, Springer, Berlin, 1979, pp. 1–9. MR 553026
G. D. Anderson, S.-L. Qiu, and M. Vuorinen, Modular equations and distortion functions, Ramanujan J. 18 (2009), no. 2, 147–169. MR 2475934, DOI 10.1007/s11139-007-9047-3
Glen D. Anderson, Mavina K. Vamanamurthy, and Matti K. Vuorinen, Conformal invariants, inequalities, and quasiconformal maps, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1997. With 1 IBM-PC floppy disk (3.5 inch; HD); A Wiley-Interscience Publication. MR 1462077
Alan F. Beardon, The geometry of discrete groups, Graduate Texts in Mathematics, vol. 91, Springer-Verlag, New York, 1983. MR 698777, DOI 10.1007/978-1-4612-1146-4
P. P. Belinskiĭ, Obshchie svoĭ stva kvazikonformnykh otobrazheniĭ, Izdat. “Nauka” Sibirsk. Otdel., Novosibirsk, 1974 (Russian). MR 0407275
C.J. Earle and L.A. Harris: Inequalities for the Carathéodory and Poincaré metrics in open unit balls, Manuscript, 2008.
F. W. Gehring and B. P. Palka, Quasiconformally homogeneous domains, J. Analyse Math. 30 (1976), 172–199. MR 437753, DOI 10.1007/BF02786713
Samuel L. Krushkal, Variational principles in the theory of quasiconformal maps, Handbook of complex analysis: geometric function theory. Vol. 2, Elsevier Sci. B. V., Amsterdam, 2005, pp. 31–98. MR 2121857, DOI 10.1016/S1874-5709(05)80006-X
J. Krzyż, On an extremal problem of F. W. Gehring, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 16 (1968), 99–101 (English, with Russian summary). MR 226002
O. Lehto and K. I. Virtanen, Quasiconformal mappings in the plane, 2nd ed., Die Grundlehren der mathematischen Wissenschaften, Band 126, Springer-Verlag, New York-Heidelberg, 1973. Translated from the German by K. W. Lucas. MR 0344463, DOI 10.1007/978-3-642-65513-5
G. J. Martin, The distortion theorem for quasiconformal mappings, Schottky’s theorem and holomorphic motions, Proc. Amer. Math. Soc. 125 (1997), no. 4, 1095–1103. MR 1363178, DOI 10.1090/S0002-9939-97-03677-0
Yu. G. Reshetnyak, Stability theorems in geometry and analysis, Mathematics and its Applications, vol. 304, Kluwer Academic Publishers Group, Dordrecht, 1994. Translated from the 1982 Russian original by N. S. Dairbekov and V. N. Dyatlov, and revised by the author; Translation edited and with a foreword by S. S. Kutateladze. MR 1326375, DOI 10.1007/978-94-015-8360-2
V. I. Semënov, Stability estimates, distortion theorems and topological properties of mappings with bounded distortion, Mat. Zametki 51 (1992), no. 5, 109–113, 159 (Russian, with Russian summary); English transl., Math. Notes 51 (1992), no. 5-6, 497–500. MR 1186539, DOI 10.1007/BF01262185
A. Yu. Solynin and M. Vuorinen, Estimates for the hyperbolic metric of the punctured plane and applications, Israel J. Math. 124 (2001), 29–60. MR 1856503, DOI 10.1007/BF02772606
Oswald Teichmüller, Ein Verschiebungssatz der quasikonformen Abbildung, Deutsche Math. 7 (1944), 336–343 (German). MR 18761
Jussi Väisälä, Lectures on $n$-dimensional quasiconformal mappings, Lecture Notes in Mathematics, Vol. 229, Springer-Verlag, Berlin-New York, 1971. MR 0454009, DOI 10.1007/BFb0061216
Matti Vuorinen, A remark on the maximal dilatation of a quasiconformal mapping, Proc. Amer. Math. Soc. 92 (1984), no. 4, 505–508. MR 760934, DOI 10.1090/S0002-9939-1984-0760934-6
Matti Vuorinen, Conformal geometry and quasiregular mappings, Lecture Notes in Mathematics, vol. 1319, Springer-Verlag, Berlin, 1988. MR 950174, DOI 10.1007/BFb0077904
Huang Xinzhong and Nak Eun Cho, On the distortion theorem for quasiconformal mappings with fixed boundary values, J. Math. Anal. Appl. 256 (2001), no. 2, 694–697. MR 1821766, DOI 10.1006/jmaa.2000.7231