Curvature properties of planar harmonic mappings
type: Item
instance of: scholarly article
title: Curvature properties of planar harmonic mappings
zbMATH Open document ID: 1051.30025
DOI: 10.1007/BF03321060
author: Peter Duren
author: Martin Chuaqui
author: Brad G. Osgood
published in: Computational Methods and Function Theory
publication date: 2004-09-28T00:00:00Z
review text: Let \(f\) be a complex-valued harmonic function defined on a domain \(D\) in the complex plane. It is well-known that a minimal surface is associated to \(f\); (for this and other basic properties of harmonic maps we refer to the recent book \textit{Harmonic mappings in the plane} by P. Duren). The authors prove a formula for the geodesic curvature of a curve on this surface in terms of \(f\). They apply this formula to establish a geometric interpretation of the Schwarzian of \(f\) (introduced in a recent paper by the same authors [J. Anal. Math. 91, 329--351 (2003; Zbl 1054.31003)]. They also derive various convexity and starlikeness results. For example they prove that if \(f\) is locally univalent in the unit disk \(\Delta\) and has a \(C^2\) extension to an arc \(I\) on \(\partial \Delta\) with unit dilatation and \(df/d\theta\neq 0\) on \(I\), then \(f(I)\) is an arc of negative curvature.
reviewed by: Dimitrios Betsakos
Mathematics Subject Classification ID: 30C99
Mathematics Subject Classification ID: 31A05
Mathematics Subject Classification ID: 53A10
zbMATH DE Number: 2104119
zbMATH Keywords: harmonic mapping
zbMATH Keywords: Schwarzian derivative
zbMATH Keywords: geodesic curvature
zbMATH Keywords: minimal surface
zbMATH Keywords: convex
zbMATH Keywords: starlike
MaRDI profile type: Publication
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Wikidata QID: Q125591780
full work available at URL: https://doi.org/10.1007/bf03321060
OpenAlex ID: W1965653208
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description: scientific article; zbMATH DE number 2104119