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 \(({{\mathcal {M}}},\textbf{g})\). In this paper we study the so-called canonical foliation on \({{\mathcal {H}}}\) introduced in [13, 22] 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 \({\textrm{tr}}\chi \) and \({\textrm{tr}}{{{\underline{\chi }}}}\) are both locally uniformly bounded. The proof of our estimates relies on a generalisation of the methods of [15,16,17] and [1, 2, 26, 32] 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 [12].