The Canonical Foliation On Null Hypersurfaces in Low Regularity
Résumé
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 [15–17] 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].