Harmonic maps between annuli on Riemann surfaces

Abstract

Let ${\rho }_{\Sigma }=h(|z{|}^2)$ be a metric in a Riemann surface $\Sigma$, where $h$ is a positive real function. Let ${\mathcal{H}}_{r_1}=\{w=f(z)\}$ be the family of univalent ${\rho }_{\Sigma }$ harmonic mapping of the Euclidean annulus $A(r_1,1):=\{z:r_1<|z|<1\}$ onto a proper annulus $A_{\Sigma }$ of the Riemann surface $\Sigma$, which is subject of some geometric restrictions. It is shown that if $A_{\Sigma }$ is fixed, then $\sup \{r_1:{\mathcal{H}}_{r_1}\ne \varnothing \}<1$. This generalizes the similar results from Euclidean case. The cases of Riemann and of hyperbolic harmonic mappings are treated in detail. Using the fact that the Gauss map of a surface with constant mean curvature (CMC) is a Riemann harmonic mapping, an application to the CMC surfaces is given (see Corollary \ref{cor}). In addition some new examples of hyperbolic and Riemann radial harmonic diffeomorphisms are given, which have inspired some new J. C. C. Nitsche type conjectures for the class of these mappings.

21 pages, to appear in Israel Journal of Mathematics

References in corpus (1)

• Harmonic mappings of an annulus, Nitsche conjecture and its generalizations

Cited by in corpus (6)

• Deformations of Annuli on Riemann surfaces with Smallest Mean Distortion
• Bi-Harmonic mappings and J. C. C. Nitsche type conjecture
• On J. C. C. Nitsche's type inequality for hyperbolic space ${\mathbf{H}}^3$
• A nonexistence result on harmonic diffeomorphisms between punctured spaces
• On J. C. C. Nitsche type inequality for annuli on Riemann surfaces
• An existence and uniqueness result for orientation-reversing harmonic diffeomorphism from ${\mathbb{H}}_{\ast }^n$ to ${\mathbb{R}}_{\ast }^n$