On J. C. C. Nitsche type inequality for annuli on Riemann surfaces

TL;DR

This paper establishes a new inequality relating the conformal modulus of annuli on Riemann surfaces to the radii of geodesic annuli under harmonic homeomorphisms, with applications to minimal surface theory.

Contribution

It proves a Nitsche-type inequality for harmonic homeomorphisms between annuli on Riemann surfaces, linking conformal modulus and geometric parameters, extending classical results.

Findings

Derived a lower bound for the ratio of radii in geodesic annuli under harmonic maps.

Connected the inequality to curvature bounds of the target surface.

Applied the result to minimal surface problems.

Abstract

Assume that $(\mathcal{N},\hbar)$ and $(\mathcal{M},\wp)$ are two Riemann surfaces with conformal metrics $\hbar$ and $\wp$. We prove that if there is a harmonic homeomorphism between an annulus $\mathcal{A}\subset \mathcal{N}$ with a conformal modulus $\mathrm{Mod}(\mathcal{A})$ and a geodesic annulus $A_\wp(p,\rho_1,\rho_2)\subset \mathcal{M}$, then we have ${\rho_2}/{\rho_1}\ge \Psi_\wp\mathrm{Mod}(\mathcal{A})^2+1,$ where $\Psi_\wp$ is a certain positive constant depending on the upper bound of Gaussian curvature of the metric $\wp$. An application for the minimal surfaces is given.