Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-10T16:11:25.575Z Has data issue: false hasContentIssue false
  • English
  • Français

NONDECREASABLE AND WEAKLY NONDECREASABLE DILATATIONS

Part of: Deformations of analytic structures Riemann surfaces Geometric function theory

Published online by Cambridge University Press:  19 September 2016

GUOWU YAO
GUOWU YAO*
Affiliation:
Department of Mathematical Sciences, Tsinghua University, Beijing 100084, PR China email gwyao@math.tsinghua.edu.cn
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Zhou et al. [‘On weakly non-decreasable quasiconformal mappings’, J. Math. Anal. Appl.386 (2012), 842–847] proved that, in a Teichmüller equivalence class, there exists an extremal quasiconformal mapping with a weakly nondecreasable dilatation. They asked whether a weakly nondecreasable dilatation is a nondecreasable dilatation. The aim of this paper is to give a negative answer to their problem. We also construct a Teichmüller class such that it contains an infinite number of weakly nondecreasable extremal representatives, only one of which is nondecreasable.

Keywords

Teichmüller spacequasiconformal mappingweakly nondecreasablepseudo nondecreasable

MSC classification

Primary: 30C75: Extremal problems for conformal and quasiconformal mappings, other methods 30C62: Quasiconformal mappings in the plane
Secondary: 30F60: Teichmüller theory 32G15: Moduli of Riemann surfaces, Teichmüller theory
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 102 , Issue 3 , June 2017 , pp. 420 - 434
Copyright
© 2016 Australian Mathematical Publishing Association Inc. 

References

Earle, C. J. and Li, Z., ‘Isometrically embedded polydisks in infinite-dimensional Teichmüller spaces’, J. Geom. Anal. 9 (1999), 51–71.Google Scholar
Fan, J., ‘On extremal quasiconformal mappings of non-landslide type’, Proc. Amer. Math. Soc. 139(8) (2011), 2729–2733.Google Scholar
Gardiner, F. P., Teichmüller Theory and Quadratic Differentials (John Wiley & Sons, New York, 1987).Google Scholar
Lakic, N., Strebel Points, Contemporary Mathematics, 211 (American Mathematical Society, Providence, RI, 1997), 417–431.Google Scholar
Lehto, O. and Virtanen, K. I., Quasiconformal Mappings in the Plane (Springer, Berlin–New York, 1973).Google Scholar
Li, Z., ‘A note on extremal quasiconformal mappings’, Sci. China Ser. A 53 (2010), 63–70.Google Scholar
Reich, E., ‘Extremal extensions from the circle to the disk’, in: Quasiconformal Mappings and Analysis, A Collection of Papers Honoring F. W. Gehring (Springer, New York–Berlin–Heidelberg, 1997), 321–335.Google Scholar
Reich, E., ‘The unique extremality counterexample’, J. Anal. Math. 75 (1998), 339–347.Google Scholar
Reich, E. and Strebel, K., ‘On quasiconformal mappings which keep the boundary points fixed’, Trans. Amer. Math. Soc. 138 (1969), 211–222.CrossRefGoogle Scholar
Reich, E. and Strebel, K., ‘Extremal quasiconformal mappings with given boundary values’, in: Contributions to Analysis, A Collection of Papers Dedicated to Lipman Bers (Academic Press, New York, 1974), 375–391.Google Scholar
Shen, Y. and Chen, J., ‘Quasiconformal mappings with non-decreasable dilatations’, Chinese J. Contemp. Math. 23 (2002), 265–276.Google Scholar
Strebel, K., Quadratic Differentials (Springer, Berlin–New York, 1984).Google Scholar
Yao, G. W., ‘Is there always an extremal Teichmüller mapping’, J. Anal. Math. 94 (2004), 363–375.Google Scholar
Yao, G. W., ‘Unique extremality, local extremality and extremal non-decreasable dilatations’, Bull. Aust. Math. Soc. 75 (2007), 321–329.Google Scholar
Yao, G. W., ‘Existence of extremal Beltrami coefficients with non-constant modulus’, Nagoya Math. J. 199 (2010), 1–14.Google Scholar
Yao, G. W., ‘Extremal Beltrami differentials of non-landslide type’, Proc. Edinb. Math. Soc. (2), to appear.Google Scholar
Yao, G. W., ‘Non-decreasable extremal Beltrami differentials of non-landslide type’. arXiv:1606.06880.Google Scholar
Zhou, Z. and Chen, J., ‘Quasiconformal mappings with non-decreasable Beltrami coefficients’, Acta Math. Sin. 46 (2003), 379–384; (in Chinese).Google Scholar
Zhou, Z., Zhang, S. and Chen, J., ‘On weakly non-decreasable quasiconformal mappings’, J. Math. Anal. Appl. 386 (2012), 842–847.Google Scholar