Nagoya Mathematical Journal

Existence of extremal Beltrami coefficients with nonconstant modulus

Guowu Yao

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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.

Article information

Nagoya Math. J., Volume 199 (2010), 1-14.

First available in Project Euclid: 14 September 2010

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 30C75: Extremal problems for conformal and quasiconformal mappings, other methods
Secondary: 30C62: Quasiconformal mappings in the plane


Yao, Guowu. Existence of extremal Beltrami coefficients with nonconstant modulus. Nagoya Math. J. 199 (2010), 1--14. doi:10.1215/00277630-2010-001.

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