Local well-posedness conjecture for the static vacuum Bartnik boundary problem

Let $(M,\bar g,\bar u)$ be an asymptotically flat, static vacuum triple, and let the geometric boundary value problem be

$$T(g,u)=(0,0,\tau,\phi),$$

where the interior equations are $-u\operatorname{Ric}_g+\nabla_g^2u=0$ and $\Delta_gu=0$, and the boundary data are $(g^\intercal,H_g)$. Here $(\tau,\phi)$ denotes prescribed boundary metric and mean curvature data, while $(\bar g^\intercal,H_{\bar g})$ are the data of the background solution. Local well-posedness conjecture. For $(\tau,\phi)$ sufficiently close to $(\bar g^\intercal,H_{\bar g})$ on $\partial M$, there exists a solution $(g,u)$ to $T(g,u)=(0,0,\tau,\phi)$ that is geometrically unique near $(\bar g,\bar u)$ and depends continuously on $(\tau,\phi)$. This local assertion is intended as a first step toward the global static vacuum extension conjecture. The paper establishes local well-posedness under its static-regularity hypotheses, but the statement as formulated for every background solution is not established in full generality.