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On extremal quasiconformal mappings of non-landslide type

Author: Fan Jinhua
Journal: Proc. Amer. Math. Soc. 139 (2011), 2729-2733
MSC (2010): Primary 30C75
Published electronically: March 24, 2011
MathSciNet review: 2801611
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $ S=\{(x,y): 0<x<1, 0<y<1\}$, and let $ f$ be a quasiconformal mapping on $ S$. It is proved that there is at least one extremal quasiconformal mapping of non-landslide type in the Teichmüller equivalence class $ [f]$. This gives a positive answer to the problem proposed by Z. Li in a recent paper.

References [Enhancements On Off] (What's this?)

  • 1. V. Bozin, N. Lakic, V. Markovic, M. Mateljevic, Unique extremality, J. Anal. Math. 75 (1998), 299-337. MR 1655836 (2000a:30045)
  • 2. R. S. Hamilton, Extremal quasiconformal mappings with prescribed boundary values, Trans. Amer. Math. Soc. 138 (1969), 399-406. MR 0245787 (39:7093)
  • 3. O. Lehto, Univalent functions and Teichmüller spaces, GTM 109, Springer-Verlag, New York, 1987. MR 867407 (88f:30073)
  • 4. Z. Li, A note on extremal quasi-conformal mappings, Sci. China Ser. A 53 (2010), 63-70. MR 2594747
  • 5. S. L. Krushkal, Extremal quasiconformal mappings, Siberian Math. J. 10 (1969), 411-418.
  • 6. E. Reich, Non-uniquely extremal quasiconformal mappings, Libertas Math. 20 (2000), 33-38. MR 1801108 (2002a:30033)
  • 7. E. Reich, The unique extremality counterexample, J. Anal. Math. 75 (1998), 339-347. MR 1655837 (99k:30037)
  • 8. E. Reich, Extremal quasiconformal mappings of the disk, in Handbook of Complex Analysis: Geometry Function Theory, vol. 1, North-Holland, 2002, pp. 75-136. MR 1966190 (2004c:30036)
  • 9. E. Reich, K. Strebel, Extremal quasiconformal mappings with given boundary values, in Contribution to Analysis, Academic Press, New York, 1974, pp. 375-392. MR 0361065 (50:13511)
  • 10. K. Strebel, Point shift differentials and extremal quasiconformal mappings, Ann. Acad. Sci. Fenn. Math. 23 (1998), 475-494. MR 1642146 (99j:30022)
  • 11. Z. Zhou, J. Chen, Z. Yang, On the extremal sets of extremal quasiconformal mappings, Sci. China Ser. A 46 (2003), 552-561. MR 2014487 (2004j:30036)

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Additional Information

Fan Jinhua
Affiliation: Department of Applied Mathematics, Nanjing University of Science and Technology, Nanjing, 210094, People’s Republic of China

Keywords: Extremal quasiconformal mapping, variability set, non-landslide type
Received by editor(s): July 1, 2010
Published electronically: March 24, 2011
Additional Notes: The author was supported by Tian-Yuan Foundation (No. 10926159) and NJUST (XKF09044).
Communicated by: Mario Bonk
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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