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 vacuum spacetime $$({{\mathcal {M}}},\textbf{g})$$ . In this paper we study so-called canonical foliation on introduced [13, 22] and show that corresponding geometry is controlled locally only terms of initial $$L^2$$ curvature flux through particular, ingoing expansions $${\textrm{tr}}\chi $$ $${\textrm{tr}}{{{\underline{\chi }}}}$$ are both uniformly bounded. The proof our estimates relies generalisation methods [15–17] [1, 2, 26, 32] where geodesic hypersurfaces studied. results paper, while independent interest, essential for spacelike-characteristic bounded theorem [12].