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The main objective of this paper is to control the geometry of null coneswith time foliation in Einstein vacuum spacetime under the assumptions of smallcurvature flux and a weaker condition on the deformation tensor for $\bT$. Weestablish a series of estimates on Ricci coefficients, which plays a crucialrole to prove the improved breakdown criterion in [12].

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