Schwarz lemma for conformal parametrization of minimal graphs in M×R

Abstract

We prove Schwarz-type lemma results for Weierstrass parameterization of the minimal disk in the Riemannian manifold $M\times R$, where M is a Riemannian surface of non-positive Gaussian curvature. The estimate is sharp, and the equality is attained if and only if the ϱ-harmonic mapping that produces the parameterization is conformal and the metric is a Euclidean metric. If the area of the minimal surface is equal to the area of the disk, then the parametrization is a contraction w.r.t. induced metric and hyperbolic metric respectively.