Kurt Strebel

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.

The Heights theorem for quadratic differentials on Riemann surfaces

1. Basic properties of quadratic differentials . . . . . . . . . . . . . 154 2. Heights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 3. A minimum norm property . . . . . . . . . . . . . . . . . . . . . 167 4. The Heights theorem and other corollaries . . . . . . . . . . . . . 172 5. Convergence of simple differentials . . . . . . . . . . . . . . . . . 176 6. Approximation by simple differentials . . . . . . . . . . . . . . . 185 7. Geometric determination of Teichmtiller mappings . . . . . . . . . 199

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 the Ends of Trajectories

The purpose of this note is to establish a certain basic property concerning in particular the “ends” of the trajectories and more generally of the geodesics of a (holomorphic) quadratic differential φ d z 2 of finite norm

On the Extremality and Unique Extremality of Certain Teichmüller Mappings

\left\| \varphi \right\| = \int {\int {\left| \varphi \right|dxdy} }

On approximation of mappings by teichmüller mappings

on an arbitrary Riemann surface R. In the remainder of this section we will briefly recall certain aspects of the geometry associated with these differentials. The theorem is stated in Sect. 2.

On the Geometry of Quadratic Differentials in the Disk

Let (fn ) be a sequence of K−qc mappings of a domain G which tends locally uniformly to a mapping f≠ const. Then it is known that f is K – qc in G. It is shown that the local dilatations Dn(z) and D(z) satisfy a.e. in G. If there is equality on a set of positive measure E ⊃ G, there exists a subsequence (fn1 ) which is a good approximation of fon E. The same method yields the convergence of arg a.e. on E