Harmonic maps between annuli on Riemann surfaces

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 $\sup\{r_1: \mathcal H_{r_1}\neq \emptyset \}<1$. 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.