Existence of extremal Beltrami coefficients with non-constant modulus

TL;DR

This paper investigates the existence of extremal Beltrami coefficients with constant modulus within the universal Teichmüller space, showing that such coefficients do not always exist when multiple extremals are present.

Contribution

It proves that extremal Beltrami coefficients with constant modulus do not necessarily exist in cases with multiple extremals, extending previous results and addressing an open problem.

Findings

Existence of extremal Beltrami coefficients with non-constant modulus confirmed.

Counterexample showing no guarantee of constant modulus extremals when multiple extremals exist.

An infinitesimal version of the problem is also established.

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.