On the Bartnik conjecture for the static vacuum Einstein equations

We prove that given any smooth metric $\gamma$ and smooth positive function $H$ on $S^{2}$, there is a constant $\lambda > 0$, depending on $(\gamma, H)$, and an asymptotically flat solution $(M, g, u)$ of the static vacuum Einstein equations on $M = {\mathbb R}^{3} \setminus B^{3}$, such that the induced metric and mean curvature of $(M, g, u)$ at $\partial M$ are given by $(\gamma, \lambda H)$. This gives a partial resolution of a conjecture of Bartnik.