The volume of conformally flat manifolds as hypersurfaces in the light-cone

In this paper, we focus on a conformally flat Riemannian manifold (Mⁿ, g) of dimension n isometrically immersed into the (n+1) -dimensional light-cone ^n+1 as a hypersurface. We compute the first and the second variational formulas on the volume of such hypersurfaces. Such a hypersurface Mⁿ is not only immersed in ^n+1 but also isometrically realized as a hypersurface of a certain null hypersurface N^n+1 in the Minkowski spacetime, which is different from ^n+1. Moreover, Mⁿ has a volume-maximizing property in N^n+1.