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 Λⁿ⁺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 Λⁿ⁺¹ but also isometrically realized as a hypersurface of a certain null hypersurface Nⁿ⁺¹ in the Minkowski spacetime, which is different from Λⁿ⁺¹. Moreover, Mⁿ has a volume-maximizing property in Nⁿ⁺¹.