Abstract
Assume that (N, ℏ) and (M, S) are two Riemann surfaces with conformal metrics ℏ and S. We prove that if there is a harmonic homeomorphism between an annulus A ⊂ N with a conformal modulus Mod(A) and a geodesic annulus AS(p, ρ1, ρ2)⊂M, then we have ρ2/ρ1 ≥ ΨSMod(A)2+ 1, where ΨS is a certain positive constant depending on the upper bound of Gaussian curvature of the metric S. An application for the minimal surfaces is given.
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