Toshiyuki Sugawa

On the degenerate Beltrami equation

We study the well-known Beltrami equation under the assumption that its measurable complex-valued coefficient μ(z) has the norm ∥μ∥∞ = 1. Sufficient conditions for the existence of a homeomorphic solution to the Beltrami equation on the Riemann sphere are given in terms of the directional dilatation coefficients of μ. A uniqueness theorem is also proved when the singular set Sing(μ) of μ is contained in a totally disconnected compact set with an additional thinness condition on Sing(μ).

Various domain constants related to uniform perfectness

This is a survey article on domain constants related to uniform perfectness. We gather comparison theorems for various domain constants, most of which are more or less known or elementary but not stated quantitatively in the literature, and some are new or improved results. Among these theorems, our main result is a comparison of the modulus and the injectivity radius of a hyperbolic Riemann surface. Its proof relies upon a comparison of extremal and hyperbolic lengths, which seems to be interesting in itself. And we include a lower estimate of the Hausdorff dimension of a compact set in the Riemann sphere by the modulus of its complement. We also discuss the variance of these domain constants under conformal, quasiconformal or Mobius maps.

NORM ESTIMATES OF THE PRE-SCHWARZIAN DERIVATIVES FOR CERTAIN CLASSES OF UNIVALENT FUNCTIONS

A sharp norm estimate will be given to the pre-Schwarzian derivatives of close-to-convex functions of specified type. In order to show the sharpness, we introduce a kind of maximal operator which may be of independent interest. We also discuss a relation between the subclasses of close-to-convex functions and the Hardy spaces.

Uniform perfectness of the limit sets of Kleinian groups

In this note, we show, in a quantitative fashion, that the limit set of a non-elementary Kleinian group is uniformly perfect if the quotient orbifold is of Lehner type, i.e., if the space of integrable holomorphic quadratic differentials on it is continuously contained in the space of (hyperbolically) bounded ones. This result covers the known case when the group is analytically finite. As applications, we present estimates of the Hausdorff dimension of the limit set and the translation lengths in the region of discontinuity for such a Kleinian group. Several examples will also be given.

Drawing bers embeddings of the teichmüller space of once-punctured tori

We present a computer-oriented method of producing pictures of Bers embeddings of the Teichmüller space of once-punctured tori. The coordinate plane is chosen in such a way that the accessory parameter is hidden in the relative position of the origin. Our algorithm consists of two steps. For each point in the coordinate plane, we first compute the corresponding monodromy representation by numerical integration along certain loops. Then we decide whether the representation is discrete by applying Jørgensens theory on the quasi-Fuchsian space of once-punctured tori.

Bers embedding of the Teichmüller space of a once-punctured torus

In this note, we present a method of computing monodromies of projective structures on a once-punctured torus. This leads to an algorithm numerically visualizing the shape of the Bers embedding of a one-dimensional Teichmüller space. As a by-product, the value of the accessory parameter of a four-times punctured sphere will be calculated in a numerical way as well as the generators of a Fuchsian group uniformizing it. Finally, we observe the relation between the Schwarzian differential equation and Heun’s differential equation in this special case.

Quasiconformal Extension of Strongly Spirallike Functions

We show that a strongly λ-spirallike function of order α can be extended to a sin(πα/2)-quasiconformal automorphism of the complex plane for -π/2 < λ < π/2 and 0 < α < 1 with ¦λ¦ < πα/2. Towards the proof we provide several geometric characterizations of a strongly λ-spirallike domain of order α. We also give a concrete form of the mapping function of the standard strongly λ-spirallike domain

Coefficient estimates of analytic endomorphisms of the unit disk fixing a point with applications to concave functions

U_{\lambda,\alpha}

On μ-conformal homeomorphisms and boundary correspondence

of order α. A key tool of the present study is the notion of λ-argument, which was developed by Y. C. Kim and the author [5].

ON A MULTI-POINT SCHWARZ-PICK LEMMA

In this note, we discuss the coefficient regions of analytic self-maps of the unit disk with a prescribed fixed point. As an application, we solve the Fekete-Szegő problem for normalized concave functions with a prescribed pole in the unit disk.