Parabolicity and Gauss map of minimal surfaces

Abstract

We prove that if a nonflat properly immersed minimal surface lies above a sublinear graph (i.e. it is contained in the region {X 3 > - (X 2 1 + X 2 2) α/2 } with 0 < α < 1) and its Gauss map is contained in an open hyperbolic subset of the sphere, then M is parabolic in the sense that bounded harmonic functions on M are determined by their boundary values. This result applies to proper minimal graphs lying above a sublinear graph.