Minimal surfaces in non-Minkowskian Randers spaces

In this paper, we investigate minimal hypersurfaces in $\mathbb{R}^n$ with respect to the Busemann—Hausdorff measure in a class of Finsler $n$-spaces ($\mathbb{R}^n,\tilde{F}_b=\tilde{\alpha}+\tilde{\beta}$), called Randers spaces, where $\tilde{\alpha} $ is the Euclidean metric and $\tilde{\beta}=b(x)dy^{n}$ 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 $(\mathbb{R}^3,\tilde{F}_b)$. 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.