Rough Solutions of Einstein Vacuum Equations in CMCSH Gauge

In this paper, we consider very rough solutions to the Cauchy problem for the Einstein vacuum equations in CMC spatial harmonic gauge, and obtain the local well-posedness result in H ^s , s > 2. The novelty of our approach lies in that, without resorting to the standard paradifferential regularization over the rough, Einstein metric g, we manage to implement the commuting vector field approach to prove Strichartz estimate for geometric wave equation ${\square_{\bf g} \phi=0}$ directly.