Hypersurfaces of Randers Spaces with Constant Mean Curvature

Abstract

Let $(\overline{M}^{n+1}, \overline{F})$ be a complete simply connected Randers space with $\overline{F}(x,Y) = \overline{a}(x,Y)+ \overline{b}(x,Y)$, where $\overline{a}(x,Y) = \sqrt{\overline{a}_{ij}(x) Y^i Y^j}$ and $\overline{b}(x,Y) = \overline{b}_i(x) Y^i$ are a Riemannian metric and a $1$-form on the smooth $(n+1)$-dimensional manifold $\overline{M}$ respectively. Assume the $1$-form $\overline{b}$ is parallel with respect to $\overline{a}$ and the sectional curvature $\overline{K}_{\overline{M}}$ of $\overline{M}$ with respect to $\overline{a}$ satisfies $\delta(n) \leq \overline{K}_{\overline{M}} \leq 1$. In this paper, we study the compact hypersurface $(M,F)$ of the Randers space $(\overline{M}^{n+1}, \overline{F})$ with constant mean curvature $|H|$ and prove that if the norm square $S$ of the second fundamental form of $(M,F)$ with respect to the Finsler metric $\overline{F}$ satisfies a certain inequality, then $S = n|H|^2$ and $M$ is the unit sphere or equality holds. In that case, we describe all $M$ that satisfy this equality, which generalizes the result of [8] from the Riemannian case to the Randers space.