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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, [email protected]
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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

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