Improved breakdown criterion for Einstein vacuum equations in CMC gauge

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Let =∪ₜ∈ [ₜ₀, ₜ*₎ Σₜ be a part of vacuum globally hyperbolic space-time (,), foliated by constant mean curvature hypersurfaces Σₜ with t₀<t*<0. We show that the foliation can be extended beyond t* if the second fundamental form k and the lapse function n satisfy ∫ₜ₀ᵗ*(kₗ∞₍Σₜ₎+ nₗ∞₍Σₜ₎) dt <∞. This improves the existing breakdown criteria for Einstein vacuum equations.