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Existence of extremal Beltrami coefficients with nonconstant modulus

Published online by Cambridge University Press:  11 January 2016

Guowu Yao
Guowu Yao*
Affiliation:
Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, People’s Republic of China, gwyao@math.tsinghua.edu.cn
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Abstract

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Suppose that [μ]T(Δ) is a point of the universal Teichmüller space T(Δ). In 1998, Božin, Lakic, Marković, and Mateljević showed that there exists μ such that μ is uniquely extremal in [μ]T(Δ) and has a nonconstant modulus. It is a natural problem whether there is always an extremal Beltrami coefficient of constant modulus in [μ]T(Δ) if [μ]T(Δ) admits infinitely many extremal Beltrami coefficients; the purpose of this paper is to show that the answer is negative. An infinitesimal version is also obtained. Extremal sets of extremal Beltrami coefficients are considered, and an open problem is proposed. The key tool of our argument is Reich’s construction theorem.

Keywords

30C7530C62
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 199 , September 2010 , pp. 1 - 14
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2010

References

[1] Božin, V., Lakic, N., Marković, V., and Mateljević, M., Unique extremality, J. Anal. Math. 75 (1998), 299–338.Google Scholar
[2] Earle, C. J. and Lakic, N., Variability sets on Riemann surfaces and forgetful maps between Teichmüller spaces, Ann. Acad. Sci. Fenn. Math. 27 (2002), 307–324.Google Scholar
[3] Fan, J. and Chen, J., On the equivalence of extremal Teichmüller mapping, Sci. China Ser. A. 52 (2009), 77–86.Google Scholar
[4] Mateljević, V. and Marković, V., The unique extremal QC mapping and uniqueness of Hanh-Banach extensions, Mat. Vesnik 48 (1996), 107–112.Google Scholar
[5] Reich, E., A criterion for unique extremality of Teichmüller mappings, Indiana Univ. Math. J. 30 (1981), 441–447.Google Scholar
[6] Reich, E., The unique extremality counterexample, J. Anal. Math. 75 (1998), 339–347.Google Scholar
[7] Reich, E., Non-uniquely extremal quasiconformal mappings, Libertas Math. 20 (2000), 33–38.Google Scholar
[8] 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
[9] Strebel, K., Point shift differentials and extremal quasiconformal mappings, Ann. Acad. Sci. Fenn. Math. 23 (1998), 475–494.Google Scholar
[10] Yao, G. W., Is there always an extremal Teichmüller mapping? J. Anal. Math. 94 (2004), 363–375.Google Scholar
[11] Yao, G. W. and Qi, Y., On the modulus of extremal Beltrami coefficients, J. Math. Kyoto Univ. 46 (2006), 235–247.Google Scholar
[12] Zhou, Z., Chen, J., and Yang, Z., On the extremal sets of extremal quasiconformal mappings, Sci. China Ser. A 46 (2003), 552–561.CrossRefGoogle Scholar