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

Abstract

Assume that $(\mathcal{N},\hslash )$ and $(\mathcal{M},\wp )$ are two Riemann surfaces with conformal metrics $\hslash$ 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.

12 pages

References in corpus (2)

• Existence of energy-minimal diffeomorphisms between doubly connected domains
• Harmonic maps between annuli on Riemann surfaces