Journal of Mathematics of Kyoto University

On the modulus of extremal Beltrami coefficients

Guowu Yao and Yi Qi

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Let $R$ be a hyperbolic Riemann surface. Suppose the Teichmüller space $T(R)$ of $R$ is infinite-dimensional. Let $\mu$ be an extremal Beltrami coefficient on $R$ and let $[\mu ]$ be the point in $T(R)$. In this note, it is shown that if $\mu$ is not uniquely extremal, then there exists an extremal Beltrami coefficient $\nu$ in $[\mu ]$ with non-constant modulus. As an application, it follows, as is well known, that there exist infinitely many geodesics between $[\mu ]$ and the base point $[0]$ in $T(R)$ if $\mu$ is non-uniquely extremal.

Article information

J. Math. Kyoto Univ., Volume 46, Number 2 (2006), 235-247.

First available in Project Euclid: 14 August 2009

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Digital Object Identifier

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; Qi, Yi. On the modulus of extremal Beltrami coefficients. J. Math. Kyoto Univ. 46 (2006), no. 2, 235--247. doi:10.1215/kjm/1250281774.

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