Low regularity well-posedness for two-dimensional deep water waves

The study of gravity-capillary water waves in two space dimensions has been an important question in mathematical fluid dynamics. By implementing the cubic modified energy method of Ifrim-Tataru in the context of gravity-capillary waves, we show that for $s>1$, the two-dimensional gravity-capillary water wave system is locally well-posed in ${\mathcal{H}}^s$.