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 $M=R^3\setminus B^3,$ 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 the conjecture of Bartnik. (paper)