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 $({\mathbb{S}}^2,\gamma ,H,P,{\omega }^{\perp })$, consisting of a metric $\gamma$, scalar functions $H$ and $P$, and a 1-form ${\omega }^{\perp }$ on the $2$-sphere ${\mathbb{S}}^2$. We construct initial data $(M,g,K)$ for the Einstein equations with boundary $\mathrm{\Sigma }\cong {\mathbb{S}}^2$, and boundary conditions for $g$ and $K$ determined by Bartnik data with $H,P$ constant and ${\omega }^{\perp }\equiv 0$. 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.