Minimal surfaces of rotation in Finsler space with a Randers metric

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=bdx3 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 get the ordinary differential equation that characterizes the minimal surfaces of rotation around the x3 axis. We prove that for every b,0≤b<1, there exists, up to homothety, a unique forward complete minimal surface of rotation. The surface is embedded, symmetric with respect to a plane perpendicular to the rotation axis and it is generated by a concave plane curve. Moreover, for every \(\) there are non complete minimal surfaces of rotation, which include explicit minimal cones.