High-frequency solutions to the constraint equations

We construct high-frequency initial data for the Einstein vacuum equations in dimension 3+1 by solving the constraint equations on ${\mathbb{R}}^3$. Our family of solutions $({\bar{g}}_{\lambda },K_{\lambda }{)}_{\lambda \in (0,1]}$ is defined through a high-frequency expansion similar to the geometric optics approach and is close in a particular sense to the data of a null dust. In order to solve the constraint equations, we use their conformal formulation and the main challenge of our proof is to adapt this method in the high-frequency context. The main application of this article is our companion article \cite{Touati2022a} where we construct high-frequency gravitational waves in generalised wave gauge.