## Nagoya Mathematical Journal

### Existence of extremal Beltrami coefficients with nonconstant modulus

Guowu Yao

#### Abstract

Suppose that $[\mu]_{T(\Delta)}$ is a point of the universal Teichmüller space $T(\Delta)$. In 1998, Božin, Lakic, Marković, and Mateljević showed that there exists $\mu$ such that $\mu$ is uniquely extremal in $[\mu]_{T(\Delta)}$ and has a nonconstant modulus. It is a natural problem whether there is always an extremal Beltrami coefficient of constant modulus in $[\mu]_{T(\Delta)}$ if $[\mu]_{T(\Delta)}$ 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

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

Dates
First available in Project Euclid: 14 September 2010

Permanent link to this document
https://projecteuclid.org/euclid.nmj/1284471568

Digital Object Identifier
doi:10.1215/00277630-2010-001

Mathematical Reviews number (MathSciNet)
MR2730409

Zentralblatt MATH identifier
1227.30021

#### Citation

Yao, Guowu. Existence of extremal Beltrami coefficients with nonconstant modulus. Nagoya Math. J. 199 (2010), 1--14. doi:10.1215/00277630-2010-001. https://projecteuclid.org/euclid.nmj/1284471568

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