PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 133, Number 9, Pages 2705–2710 S 0002-9939(05)07817-2 Article electronically published on March 22, 2005 ELLIPSES, NEAR ELLIPSES, AND HARMONIC M ¨ OBIUS TRANSFORMATIONS MARTIN CHUAQUI, PETER DUREN, AND BRAD OSGOOD (Communicated by Juha M. Heinonen) Abstract. It is shown that an analytic function taking circles to ellipses must be a M¨obius transformation. It then follows that a harmonic mapping taking circles to ellipses is a harmonic M¨obius transformation. Analytic M¨ obius 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 M¨ obius transformations as a general- ization of M¨ obius 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 M¨ obius 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 M¨ obius transformations We begin with a very brief review of the definition and properties of harmonic mappings and harmonic M¨ obius 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 3 . 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. Key words and phrases. Harmonic mapping, Schwarzian derivative, harmonic M¨obius trans- formation, circles, ellipses. The first author was supported by Fondecyt Grant # 1030589. c 2005 American Mathematical Society 2705 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use