Bounded $L^2$-curvature conjecture for the Einstein scalar field equations

Let $(\Sigma,g,k,u_{0},u_{1})$ be a smooth initial data set for the Einstein scalar field equations on a three-dimensional manifold $\Sigma$, satisfying the constraint equations. Here $g$ is a Riemannian metric, $k$ is a symmetric $2$-tensor, and $u_{0},u_{1}$ are functions on $\Sigma$ representing the initial scalar field and its normal derivative.

Bounded $L^2$-curvature conjecture. The Einstein scalar field equations admit local Cauchy developments for initial data sets $(\Sigma,g,k,u_{0},u_{1})$ with locally finite $L^{2}$-curvature, locally finite $L^{2}$-norm of the first covariant derivatives of $k$, locally finite $L^{2}$-norm of the covariant derivatives up to second order of $u_{0}$, and locally finite $L^{2}$-norm of the covariant derivatives up to first order of $u_{1}$.

This conjecture extends the bounded $L^{2}$-curvature local-development result for the Einstein vacuum equations to the Einstein scalar field system. The corresponding vacuum conjecture was solved by Klainerman, Rodnianski and Szeftel, while the scalar-field assertion remains proposed in the source.