Existence of extremal Beltrami coefficients with nonconstant modulus

Abstract

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

On the equivalence of extremal Teichmüller mapping

“…Following Theorems 1 and 2, we answer some problems proposed in [11,12] by Yao. In the next section, we introduce the problems.…”

“…From the proof of Reich's construction theorem, Yao found that the requirement of Δ\A being doubly connected is not essential and he proved the following Construction theorem [12] .…”

“…, m) be m Jordan domains such that A i ∩ A j = ∅ (i = j). By the above construction theorem, Yao [12] constructed a Beltrami cofficient as follows:…”

“…In [12], Yao claimed that [η] T (Δ) contains infinitely many extremal Beltrami coefficients but no extremal Beltrami coefficients with constant modulus, To such η(z), Yao also proved that [η] B(Δ) contains infinitely many extremal Beltrami coefficients but no extremal Beltrami coefficients with constant modulus. Let η(z) be defined as in (5.2).…”

“…The proofs of Theorems 1 and 2 are given in Section 3 and Section 4 respectively. In Section 5, we state some results in [12] and give answers to the problems.…”