Parabolicity and Gauss map of minimal surfaces

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(1)(2) + x(2)(2))(alpha/2)} with 0 < alpha < 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.