On the radius of injectivity of null hypersurfaces

We investigate the regularity of past (future) boundaries of points in regular, Einstein vacuum spacetimes. We provide conditions, expressed in terms of a space-like foliation and which imply, in particular, uniform $L^2$ bounds for the curvature tensor, sufficient to ensure the local nondegeneracy of these boundaries. More precisely we provide a uniform lower bound on the radius of injectivity of the null boundaries $\mathcal{N}^{\pm}(p)$ of the causal past (future) sets $\mathcal{J}^{\pm}(p)$. Such lower bounds are essential in understanding the causal structure and the related propagation properties of solutions to the Einstein equations. They are particularly important in construction of an effective Kirchoff-Sobolev type parametrix for solutions of wave equations on $\mathbf{M}$. Such parametrices are used by the authors in a forthcoming paper to prove a large data break-down criterion for solutions of the Einstein vacuum equations.