  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.
