On Ricci coefficients of null hypersurfaces with time foliation in Einstein vacuum space-time: part I

The main objective of this paper is to control the geometry of null cones with time foliation in Einstein vacuum spacetime under the assumptions of small curvature flux and a weak condition on the deformation tensor for the future directed unit normal T to each leaf. We establish a series of estimates on Ricci coefficients, which play a crucial role to prove the improved breakdown criterion in Wang (Commun. Pure Appl. Math. 65(1):0021–0076, 2012).