A Bernstein type theorem on a Randers space

We consider Finsler spaces with a Randers metric F=+, on the three-dimensional real vector space, where is the Euclidean metric and is a 1-form with norm b,0 b1. By using the notion of mean curvature for immersions in Finsler spaces, introduced by Z. Shen, we obtain the partial differential equation that characterizes the minimal surfaces which are graphs of functions. For each b, 0 b1/, we prove that it is an elliptic equation of mean curvature type. Then the Bernstein type theorem and other properties, such as the nonexistence of isolated singularities, of the solutions of this equation follow from the theory developped by L. Simon. For b 1/, the differential equation is not elliptic. Moreover, for every b, 1/b1 we provide solutions, which describe minimal cones, with an isolated singularity at the origin.Partially supported by CAPES/PROCAD.Partially supported by NSF grant DMS-0072242.Partially supported by CNPq and CAPES/PROCAD.