Existence of extremal Beltrami coefficients with non-constant modulus

Abstract

Suppose $[\mu]_{T(\Delta)}$ is a point of the universal Teichm\"uller space $T(\Delta)$. In 1998, it was shown by Bo\v{z}in et al. that there exists $\mu$ such that $\mu$ has non-constant modulus and is uniquely extremal in $[\mu]_{T(\Delta)}$. It is a natural problem whether there is always an extremal Beltrmai coefficient of constant modulus in $[\mu]_{T(\Delta)}$ if $[\mu]_{T(\Delta)}$ admits more than one extremal Beltrami coefficient. 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.