This chapter deals with harmonic mappings of the unit disk onto convex regions. The simplest examples, the harmonic self-mappings of the disk, are singled out for detailed treatment in Chapter 4. The present chapter will focus on two important structural properties of convex mappings. The first is the celebrated Radó–Kneser–Choquet theorem, which constructs a harmonic mapping of the disk onto any bounded convex domain, with prescribed boundary correspondence. The second is the “shear construction” of a harmonic mapping with prescribed dilatation onto a domain convex in a given direction. This leads to an analytic description of convex mappings, which has various applications.

The Radó–Kneser–Choquet Theorem

Let Ω ⊂ ℂ be a domain bounded by a Jordan curve Γ. Each homeomorphism of the unit circle onto Γ has a unique harmonic extension to the unit disk, defined by the Poisson integral formula. The values of this harmonic extension must lie in the closed convex hull of Ω in view of the “averaging” property of the Poisson integral. It is a remarkable fact that if Ω is convex, this harmonic extension is always univalent and it maps the disk harmonically onto Ω.

This theorem was first stated in 1926 by Tibor Radó [1], who posed it as a problem in the Jahresberichte. Helmut Kneser [1] then supplied a brief but elegant proof. A period of almost 20 years elapsed before Gustave Choquet [1], apparently unaware of Kneser's note, rediscovered the result and gave a detailed proof that has some features in common with Kneser's but is not the same. In fact, the two approaches allow the theorem to be generalized in different directions. We shall present both proofs, beginning with Kneser's.

Harmonic mappings in the plane are univalent complex-valued harmonic functions whose real and imaginary parts are not necessarily conjugate. In other words, the Cauchy–Riemann equations need not be satisfied, so the functions need not be analytic. Although harmonic mappings are natural generalizations of conformal mappings, they were studied originally by differential geometers because of their natural role in parametrizing minimal surfaces. Only in the mid-1980s did harmonic mappings begin to attract widespread interest among complex analysts. The catalyst was a landmark paper by James Clunie and Terry Sheil-Small in 1984, pointing out that many of the classical results for conformal mappings have clear analogues for harmonic mappings. Since that time the subject has developed rapidly, although a number of basic problems remain unresolved. This book is an attempt to make this beautiful material accessible to a wider mathematical public.

Most of the book concerns harmonic mappings in the plane, but there are occasional excursions into higher dimensions, if only to provide counterexamples. As a general rule, the rich structure of theory in the plane does not extend to higher-dimensional space. In many instances, the properties of analytic univalent functions serve as models for generalizations to harmonic mappings, but other results are peculiar to analytic functions and do not extend to more general harmonic mappings. On the other hand, some results for harmonic mappings have no counterpart for conformal mappings. This is particularly true of the connections with minimal surfaces, which are developed in the final two chapters.