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

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

Please note special location and time. The commuting vector fields approach, devised for Strichartz estimates by Klainerman, was employed 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 him and Rodnianski. Via this approach they obtained the local well-posedness in $H^s$ with $s>2$ for $(1+3)$ vacuum Einstein equations, by taking advantage of the vanishing Ricci curvature. The sharp, $H^{2+\epsilon}$, local well-posedness result for general quasilinear wave equation was achieved by Smith and Tataru by constructing a parametrix using wave packets. Using the vector fields approach, one has to face the major hurdle caused by the Ricci tensor of the metric for the quasi-linear wave equations. This posed a question that if the geometric approach can provide the sharp result for the non-geometric equations. I will present my recent work, which proves the sharp local well-posedness of general quasilinear wave equation in ${\Bbb R}^{1+3}$ by a vector field approach, based on geometric normalization and new observations on the mass aspect functions.