Existence of extremal Beltrami coefficients with nonconstant modulus

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.