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The commuting vector fields approach, devised for strichartz estimates in[13], was developed for proving the local well-posedness in the Sobolev spaces$H^s$ with $s>2+\frac{2-\sqrt{3}}{2}$ for general quasi-linear wave equation in${\mathbb R}^{1+3}$ by Klainerman and Rodnianski. Via this approach theyobtained the local well-posedness in $H^s$ with $s>2$ for $(1+3)$ vacuumEinstein equations, by taking advantage of the vanishing Ricci curvature. Thesharp, $H^{2+\epsilon}$, local well-posedness result for general quasilinearwave equation was achieved by Smith and Tataru by constructing a parametrixusing wave packets. Using the vector fields approach, one has to face the majorhurdle caused by the Ricci tensor of the metric for the quasi-linear waveequations. This posed a question that if the geometric approach can provide thesharp result for the non-geometric equations. In this paper, based on geometricnormalization and new observations on the mass aspect function, we prove thesharp local well-posedness of general quasilinear wave equation in ${\BbbR}^{1+3}$ by a vector field approach.

A geometric approach for sharp Local well-posedness of quasilinear wave equations