The volume of marginally trapped submanifolds in a Lorentzian manifold satisfying the null energy condition

In this paper, we focus on a marginally trapped submanifold $f:{\mathrm{\Sigma }}^n\to M_1^{n+2}$ in a Lorentzian manifold $M_1^{n+2}$. We show that $f$ lies in a certain null hypersurface ${\mathcal{N}}_f^{n+1}$ in $M_1^{n+2}$ and $f$ has a locally volume-maximizing property in ${\mathcal{N}}_f^{n+1}$ if $M_1^{n+2}$ satisfies the null energy condition.