Curvature properties of planar harmonic mappings
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.
scientific article; zbMATH DE number 2104119
scientific article