Mean Curvature Flow of Spacelike Hypersurfaces near Null Initial Data

摘要

We prove an interior estimate for the gradient function of space-like hypersurfaces which move by mean curvature in a Loreritzian manifold. This estimate depends only on a time function 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 geodesies which extend to its boundary.