Harmonic maps between annuli on Riemann surfaces

TL;DR

This paper investigates harmonic maps between annuli on Riemann surfaces with various metrics, establishing bounds on the annulus radii and exploring applications to constant mean curvature surfaces, including new examples and conjectures.

Contribution

It generalizes known Euclidean results to Riemann surfaces, analyzes harmonic mappings with different metrics, and connects these to CMC surface theory with new examples and conjectures.

Findings

Bound on the maximal annulus radius for harmonic maps

Detailed analysis of Riemann and hyperbolic harmonic mappings

New examples of hyperbolic and Riemann radial harmonic diffeomorphisms

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