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

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

2012-04-24

arxiv.org/abs/1204.5419v1

Mathematics

Differential Geometry

12 pages

Scientific paper

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.

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