Ellipses, near Ellipses, and Harmonic Möbius Transformations

It is shown that an analytic function taking circles to ellipses must be a Möbius transformation. It then follows that a harmonic mapping taking circles to ellipses is a harmonic Möbius transformation. Analytic Möbius transformations take circles to circles. This is their most basic, most celebrated geometric property. We add the adjective ‘analytic’ because in a previous paper [1] we introduced harmonic Möbius transformations as a generalization of Möbius transformations to harmonic mappings. Their basic geometric property, the only one we know so far, is that they take circles to ellipses. In this paper we consider the converse question. We shall show that a harmonic mapping that takes circles to ellipses must be a harmonic Möbius transformation. We also have some comments on the situation for analytic functions; in fact, we need a similar result for analytic functions to deal with the harmonic case. 1. Harmonic mappings and harmonic Möbius transformations We begin with a very brief review of the definition and properties of harmonic mappings and harmonic Möbius transformations, followed by a statement of our main result. A harmonic, complex-valued function f defined on a simply connected domain can be written in the form f = h+g, where h and g are analytic. When f is locally univalent and sense-preserving one has h′(z) = 0 and the analytic function ω = g′/h′, called the (second) complex dilatation of f , satisfies |ω(z)| < 1. In this paper we will always assume that a harmonic function f is locally univalent and sense-preserving, and we refer to f as a harmonic mapping. On any neighborhood where ω is not zero or has zeros of even order, f lifts to a mapping whose image is a minimal surface in R. The metric of the surface has the form ρ|dz| where ρ = |h′|+ |g′| and the curvature is K = − |ω ′|2 |h′||g′|(1 + |ω|2) . We refer to [2] for further background. Received by the editors January 22, 2004 and, in revised form, April 29, 2004. 2000 Mathematics Subject Classification. Primary 30C99; Secondary 31A05.