The canonical foliation on null hypersurfaces in low regularity

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 $\mathrm{t}\mathrm{r}\chi$ and $$\mathrm{t}\mathrm{r}$ \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.