IUMJ

Title: Hereditary convexity for harmonic homeomorphisms

Authors: Ngin-tee Koh

Issue: Volume 64 (2015), Issue 1, 231-243

Abstract:

We study hereditary properties of convexity for planar harmonic homeomorphisms on a disk and an annulus. A noteworthy class of examples with the hereditary property arises from energy-minimal diffeomorphisms of an annulus, whose existence was established in [T. Iwaniec, N.T. Koh, L.V. Kovalev, and J. Onninen, \textit{Existence of energy-minimal diffeomorphisms between doubly connected domains}, Invent. Math. \textbf{186} (2011), no. 3, 667-707], [D. Kalaj, \textit{Energy-minimal diffeomorphisms between doubly connected Riemann surfaces}, Calc. Var. Partial Differential Equations \textbf{51} (2014), no. 1--2, 465--494]. An extension of a result by Hengartner and Schober [W. Hengartner and G. Schober, \textit{Harmonic mappings with given dilatation}, J. London Math. Soc. (2) \textbf{33} (1986), no. 3, 473--483] to an annulus is used to deduce the boundary behavior of a harmonic mapping from an annulus into a doubly-connected region bounded by two convex Jordan curves.