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

TL;DR

This paper investigates conformally flat manifolds immersed as hypersurfaces in the light-cone, deriving variational formulas for their volume and revealing volume-maximizing properties within Minkowski spacetime.

Contribution

It introduces a novel analysis of conformally flat hypersurfaces in the light-cone, including variational formulas and volume-maximizing characteristics in null hypersurfaces.

Findings

Computed first and second variational formulas for volume.

Established volume-maximizing property in Minkowski spacetime.

Connected hypersurfaces in the light-cone with null hypersurfaces in Minkowski space.

Abstract

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