The canonical foliation on null hypersurfaces in low regularity

TL;DR

This paper establishes local control of the geometry of null hypersurfaces in vacuum spacetimes with low regularity, using initial data and curvature flux, advancing understanding of null foliation geometry.

Contribution

It demonstrates that the canonical foliation geometry on null hypersurfaces can be controlled solely by initial data and curvature flux in low regularity settings.

Findings

Null expansions are uniformly bounded.

Geometry controlled by initial data and curvature flux.

Extends methods of Klainerman-Rodnianski and others.

Abstract

Let $\mathcal{H}$ denote the future outgoing null hypersurface emanating from a spacelike 2-sphere $S$ in a vacuum spacetime $(\mathcal{M},\mathbf{g})$. In this paper we study the so-called canonical foliation on $\mathcal{H}$ introduced by Klainerman and Nicol\`o and show that the corresponding geometry is controlled locally only in terms of the initial geometry on $S$ and the $L^2$ curvature flux through $\mathcal{H}$. In particular, we show that the ingoing and outgoing null expansions $\mathrm{tr} \chi$ and $\mathrm{tr} \underline{\chi}$ are both locally uniformly bounded. The proof of our estimates relies on a generalisation of the methods of Klainerman and Rodnianski, and Alexakis, Shao and Wang where the geodesic foliation on null hypersurfaces $\mathcal{H}$ is studied. The results of this paper, while of independent interest, are essential for the proof of the spacelike-characteristic bounded $L^2$ curvature theorem by Czimek and Graf.