Extensions of spacetime Bartnik data and estimates for the Bartnik mass outside of time-symmetry

Bartnik's quasi-local mass is a functional on Bartnik data $(\mathbb S^2,γ,H,P,ω^\perp)$, consisting of a metric $γ$, scalar functions $H$ and $P$, and a 1-form $ω^\perp$ on the $2$-sphere $\mathbb S^2$. We construct initial data $(M,g,K)$ for the Einstein equations with boundary $Σ\cong\mathbb S^2$, and boundary conditions for $g$ and $K$ determined by Bartnik data with $H,P$ constant and $ω^\perp\equiv0$. Furthermore this initial data agrees with spherically symmetric initial data for a Schwarzschild spacetime outside of a compact set with controlled mass. As an application, we obtain estimates for the Bartnik mass for such Bartnik data, outside of the time-symmetric setting. We also construct initial data on the cylinder $\mathbb S^2\times[0,1]$ connecting this same class of Bartnik data to time-symmetric data so that estimates for the Bartnik mass outside of time-symmetry can be obtained from prior estimates for time-symmetric data.