Mean curvature flow of spacelike hypersurfaces near null initial data

We prove an interior estimate for the gradient function of space-like hypersurfaces which move by mean curvature in a Lorentzian manifold. This estimate depends only on a time fraction which measures how far the hypersurfaces are from being null. As a consequence, we show that under mean curvature flow a weakly spacelike initial hypersurface instantaneously becomes smooth and strictly spacelike except along null geodesics which extend to its boundary.