A class of critical surfaces in a Finsler space under the volume preserving variation

Let $(\mathbb{R}^3,\tilde{F}_b)$ be the 3-dimensional Randers space with the metric \[\tilde{F}_b=\sqrt{(dx^1)^2+(dx^2)^2+(dx^3)^2}+b dx^3,\] where $0\leq b＜1$ is a constant. In this paper, we study the critical surfaces under the volume preserving variation in $(\mathbb{R}^3,\tilde{F}_b)$ under the Busemann—Hausdorff measure. We introduce a quantity $H_\sigma=\operatorname{const.}$ to characterize such surfaces which are called the constant mean curvature surfaces. Similar to Delaunay's famous work [10], we give a complete classification of CMC surfaces rotating around the $x^3$-axis in the 3-dimensional Randers space with the Busemann—Hausdorff measure, which reduces to the classical classification of Delaunay's CMC surfaces in $\mathbb{R}^3$ when $b=0$. The method developed in this paper may be applied to find the CMC surfaces under the Holmes—Thompson measure in the $(\alpha,\beta)$-spaces.