3 - Harmonic Mappings onto Convex Regions

Summary

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.