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

TL;DR

This paper investigates the geometric properties of marginally trapped submanifolds in Lorentzian manifolds satisfying the null energy condition, revealing their containment in null hypersurfaces and their volume-maximizing behavior.

Contribution

It demonstrates that such submanifolds lie in null hypersurfaces and possess a local volume-maximizing property under the null energy condition, advancing understanding of their geometric structure.

Findings

Submanifolds lie in specific null hypersurfaces.

They exhibit a local volume-maximizing property.

Results depend on the null energy condition.

Abstract

In this paper, we focus on a marginally trapped submanifold $f:\Sigma^n\to M^{n+2}_1$ in a Lorentzian manifold $M^{n+2}_1$. We show that $f$ lies in a certain null hypersurface $\mathcal{N}_f^{n+1}$ in $M^{n+2}_1$ and $f$ has a locally volume-maximizing property in $\mathcal{N}^{n+1}_f$ if $M^{n+2}_1$ satisfies the null energy condition.