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.
Given a quasi-symmetric self-homeomorphism h of the unit circle S l , let Q(h) be the set of all quasiconformal mappings with the boundary correspondence h. In this paper, it is shown that there exists certain quasi-symmetric homeomorphism h, such that Q(h) satisfies either of the conditions,(1) Q{h) admits a quasiconformal mapping that is both uniquely locallyextremal and uniquely extremal-non-decreasable instead of being uniquely extremal; (2) Q(h) contains infinitely many quasiconformal mappings each of which has an extremal non-decreasable dilatation. An infinitesimal version of this result is also obtained.
Recently, the first author and Y. Wang proved that $f\colon \rn\to\rn$ (n ≥ 2) is a Möbius transformation if and only if f is a non-degenerate circle-preserving map. In this paper, we will further the result to show that f is a Möbius transformation if and only if f is a non-degenerate r–dimensional sphere-preserving map. The versions for the Euclidean and hyperbolic cases are also obtained. These results make no surjectivity or injectivity or even continuity assumptions on f. Moreover, certain degenerate sphere-preserving maps are given, which completes the characterizations of sphere-preserving maps.
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