A Bernstein type theorem on a Randers space

Abstract

We consider Finsler spaces with a Randers metric F=α+β, on the three-dimensional real vector space, where α is the Euclidean metric and β is a 1-form with norm b,0≤ b≤1. 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≤ b≤1/Open image in new window, 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/Open image in new window, the differential equation is not elliptic. Moreover, for every b, 1/Open image in new window≤b≤1 we provide solutions, which describe minimal cones, with an isolated singularity at the origin.