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

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.