Edgar Reich

Extremal Quasiconformal Mappings with Given Boundary Values

Publisher Summary This chapter discusses the quasiconformal self mappings f= fx of the unit disc E: |z| <1. The superscript x denotes the complex dilatation of the mapping f. By continuation, f induces a homeomorphism of the boundary ∂E onto itself. The chapter presents a proof of Hamiltons theorem for the unit disc, based on the length-area method. The quadratic differentials φn determined by the extremal quasiconformal mappings of the unit disc with n distinguished boundary points form a maximizing sequence for this integral. The chapter discusses an open problem regarding whether the condition (*) is also sufficient, in other words, whether the converse of Hamiltons theorem is true. The proof presented in the chapter makes use only of the length-area method and well-known properties of quadratic differentials with finite norm.

On quasiconformal mappings which keep the boundary points fixed

Introduction. From a variational point of view it is natural to consider quasiconformal selfmappings of a domain which are equal to the identity on the boundary. We say that a function / in the unit disk U= {z I Izl O such that k(+/11I) E E The last section of the present work is devoted to this question. A measurable and bounded complex valued function v in U is said to belong to the class N(Ahlfors [1]) if fu v(z)g(z) dx dy=O for every holomorphic function g in U with 11 g 1 0(2).

Extremal plane quasiconformal mappings with given boundary values

Publisher Summary This chapter discusses the quasiconformal self mappings f = f x of the unit disc E: |z| x denotes the complex dilatation of the mapping f . By continuation, f induces a homeomorphism of the boundary ∂E onto itself. The chapter presents a proof of Hamiltons theorem for the unit disc, based on the length-area method. The quadratic differentials φ n determined by the extremal quasiconformal mappings of the unit disc with n distinguished boundary points form a maximizing sequence for this integral. The chapter discusses an open problem regarding whether the condition (*) is also sufficient, in other words, whether the converse of Hamiltons theorem is true. The proof presented in the chapter makes use only of the length-area method and well-known properties of quadratic differentials with finite norm.

ESTIMATES FOR THE TRANSFINITE DIAMETER OF A CONTINUUM

Abstract : Some estimates for the transfinite diameter of a continuum are derived in terms of elementary geometric quantities connected with it. The main intention is methodological. Variational methods are applied, and the corresponding extremum continua are described by a functional differential equation. In all cases the determination of the extremum continuum is based on rather simple geometric arguments. It is intended that the arguments used should indicate more general methods of solution which could be applied in the calculus of variations for conformal mappings. (Author)

On the Gerstenhaber-Rauch principle

Supposef*(z) is aK*-qc self-homeomorphism of the unit diskU, whereK* is the minimum possible value among all qc mappings ofU with the same boundary values asf*. It is known thatK* can be calculated by a variational principle involving mappings ofU harmonic with respect to admissible weight functions. We examine the weight functions that correspond to the case when the extremum for the variational principle is attained, and characterize the corresponding mappingsf*.

On teichmüller mappings of the disk

Let be regular in the unit disk U and satisfy the growth condition . In a sense made precise in Theorem 1, it is shown that if 0<t0≤t<1, then the L1 norm of over the set where is uniformly small if A 0 is sufficiently large. The principal objective is to derive Theorem 2: If f is a quasiconformal mapping of U, , then f is uniquely extremal among the class of mappings g of U, with .

Extremal Extensions from the Circle to the Disk

Let\( \Delta = \{ |z|{\rm{ }} < {\rm{1\} , }}\Gamma {\rm{ = \{ |}}z{\rm{| = 1\} , }}\partial {\rm{ = (}}\partial _x - i\partial _y )/2, \bar \partial = (\partial _x + i\partial _y )/2 \) In this exposition we shall be concerned with two kinds of extension problems from \( \Gamma to\Delta \cup \Gamma \), namely, firstly, the least non-analytic (abbreviated LNA) extensions of continuous complex-valued boundary values, and, secondly, the extremal quasiconformal (abbreviated EQC) extensions of homeomorphic boundary values. We formulate the first problem as follows. Let \( g(z),z \in \Gamma \), be a continuous complex-valued function. It will often be convenient to normalize g by the conditions

A theorem of fehlmann-type for extensions with bounded -derivative

Re[\bar zg(z)] = 0, z \in \Gamma ,