Mean Curvature Flow of Spacelike Hypersurfaces near Null Initial Data

We prove an interior estimate for the gradient function of spacelike hypersurfaces which move by mean curvature in a Lorentzian 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 geodesics which extend to its boundary.

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Published 1 January 2003