Hypersurfaces of Randers Spaces with Constant Mean Curvature

Let ((M) over bar (n+1), (F) over bar) be a complete simply connected Randers space with (F) over bar (x, Y) = (a) over bar (x, Y) + (b) over bar (x, Y), where (a) over bar (x, Y) = root(a) over bar ij(x)(YYj)-Y-i and (b) over bar (x, Y) = (b) over bar (i)(x)Y-i are a Riemannian metric and a 1-form on the smooth (n + 1)-dimensional manifold (M) over bar respectively. Assume the 1-form (b) over bar is parallel with respect to (a) over bar and the sectional curvature (K) over bar ((M) over bar) of (M) over bar with respect to (a) over bar satisfies delta(n) <= (K) over bar ((M) over bar) <= 1. In this paper, we study the compact hypersurface (M, F) of the Randers space ((M) over bar (n+1), (F) over bar) with constant mean curvature vertical bar H vertical bar and prove that if the norm square S of the second fundamental form of (M, F) with respect to the Finsler metric (F) over bar satisfies a certain inequality, then S = n vertical bar H vertical bar(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.