VMO-Teichmüller space on the real line

An increasing homeomorphism \(h\) on the real line \(\mathbb{R}\) is said to be strongly symmetric if it can be extended to a quasiconformal homeomorphism of the upper half plane \(\mathbb{U}\) onto itself whose Beltrami coefficient \(\mu\) induces a vanishing Carleson measure \(|\mu(z)|^2/y\,dx\,dy\) on \(\mathbb{U}\). We will deal with the class of strongly symmetric homeomorphisms on the real line and its Teichmüller space, which we call the VMO-Teichmüller space. In particular, we will show that if \(h\) is strongly symmetric on the real line, then it is strongly quasisymmetric such that \(\log h'\) is a VMO function. This improves some classical results of Carleson (1967) and Anderson-Becker-Lesley (1988) on the problem about the local absolute continuity of a quasisymmetric homeomorphism in terms of the Beltrami coefficient of a quasiconformal extension. We will also discuss various models of the VMO-Teichmüller space and endow it with a complex Banach manifold structure via the standard Bers embedding.

The Harmonic Mapping Whose Hopf Differential Is a Constant

Suppose that h(z) is a harmonic mapping from the unit disk D to itself with respect to the hyperbolic metric. If the Hopf differential of h(z) is a constant c>0, the Beltrami coefficient μ(z) of h(z) is radially symmetric and takes the maximum at z=0. Furthermore, the mapping γ:c→μ(0) is increasing and gives a homeomorphism from (0,+∞) to (0,1).

Asymptotic conformality of the barycentric extension of quasiconformal maps

We first remark that the complex dilatation of a quasiconformal homeomorphism of a hyperbolic Riemann surface R obtained by the barycentric extension due to Douady-Earle vanishes at any cusp of R. Then we give a new proof, without using the Bers embedding, of a fact that the quasiconformal homeomorphism obtained by the barycentric extension from an integrable Beltrami coefficient on R is asymptotically conformal if R satisfies a certain geometric condition.

Beltrami Equation with Coefficient in Sobolev and Besov Spaces

Our goal in this work is to present some function spaces on the complex plane ℂ, X(ℂ), for which the quasiregular solutions of the Beltrami equation, ∂̄ f (z) = μ(z)∂ f (z), have first derivatives locally in X(ℂ), provided that the Beltrami coefficient μ belongs to X(ℂ).

ON KELLOGG'S THEOREM FOR QUASICONFORMAL MAPPINGS

We give some extensions of classical results of Kellogg and Warschawski to a class of quasiconformal (q.c.) mappings. Among the other results we prove that a q.c. mappingf, between two planar domains with smoothC1,αboundaries, together with its inverse mappingf−1, isC1,αup to the boundary if and only if the Beltrami coefficient μfis uniformly α Hölder continuous (0 < α < 1).

Existence of extremal Beltrami coefficients with nonconstant modulus

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.

Existence of extremal Beltrami coefficients with nonconstant modulus

Suppose that [μ]T(Δ)is a point of the universal Teichmüller spaceT(Δ). In 1998, Božin, Lakic, Marković, and Mateljević showed that there existsμsuch thatμis 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.