A note on inverse mean curvatrue flow in cosmological spacetimes

TL;DR

This paper extends previous results on inverse mean curvature flow in cosmological spacetimes by relaxing some geometric assumptions, showing the flow still produces a foliation under weaker conditions.

Contribution

It generalizes Gerhardt's earlier work by removing the mean curvature barrier assumption and weakening the energy condition needed for the flow's long-time existence.

Findings

Flow provides a foliation of the future of the initial hypersurface.

Longtime existence persists under weaker geometric conditions.

Results are applicable to physically relevant energy conditions.

Abstract

In [8] Gerhardt proves longtime existence for the inverse mean curvature flow in globally hyperbolic Lorentzian manifolds with compact Cauchy hypersurface, which satisfy three main structural assumptions: a strong volume decay condition, a mean curvature barrier condition and the timelike convergence condition. Furthermore, it is shown in [8] that the leaves of the inverse mean curvature flow provide a foliation of the future of the initial hypersurface. We show that this result persists, if we generalize the setting by leaving the mean curvature barrier assumption out. For initial hypersurfaces with sufficiently large mean curvature we can weaken the timelike convergence condition to a physically relevant energy condition.