Minimal surfaces in non-Minkowskian Randers spaces

Abstract

In this paper, we investigate minimal hypersurfaces in ℝ𝑛 with respect to the Busemann—Hausdorff measure in a class of Finsler 𝑛-spaces (ℝ𝑛,˜𝐹𝑏 =˜𝛼 +˜𝛽), called Randers spaces, where ˜𝛼 is the Euclidean metric and ˜𝛽 =𝑏(𝑥)𝑑𝑦𝑛 is a controlled one-form. We emphasize the fact that 𝐹 is non-Minkowskian, since 𝑏 =𝑏(𝑥) is a non-constant function of 𝑥, which is allowed here. We particularly examine graphs defined on the 𝑥𝑦-plane that are invariant under one-dimensional isometry groups of (ℝ3,˜𝐹𝑏). By reducing the minimal graph equation to an ordinary differential equation (ODE), we obtain a new class of explicit examples of minimal surfaces in Finsler geometry.