Ellipticity of Bartnik boundary data for stationary vacuum spacetimes

We establish a moduli space $\mathbb{E}$ of stationary vacuum metrics in a spacetime, and set up a well-defined boundary map $\mathrm{\Pi }$ in $\mathbb{E}$, assigning a metric class with its Bartnik boundary data. Furthermore, we prove the boundary map $\mathrm{\Pi }$ is Fredholm by showing that the stationary vacuum equations (combined with proper gauge terms) and the Bartnik boundary conditions form an elliptic boundary value problem. As an application, we show that the Bartnik boundary data near the standard flat boundary data admits a unique (up to diffeomorphism) stationary vacuum extension locally.