Home
Hostname: page-component-99c86f546-t82dr Total loading time: 0.321 Render date: 2021-12-02T08:21:26.846Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": true, "newCiteModal": false, "newCitedByModal": true, "newEcommerce": true, "newUsageEvents": true }

# 3 - Harmonic Mappings onto Convex Regions

Published online by Cambridge University Press:  15 September 2009

## 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.

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.

Type
Chapter
Information
Publisher: Cambridge University Press
Print publication year: 2004

## Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

### Purchase

Buy print or eBook[Opens in a new window]

# Send book to Kindle

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

# Send book to Dropbox

To send content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about sending content to Dropbox.

Available formats
×

# Send book to Google Drive

To send content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about sending content to Google Drive.

Available formats
×