On the Bartnik extension problem for the static vacuum Einstein equations

We develop a framework for understanding the existence of asymptotically flat solutions to the static vacuum Einstein equations onM = ℝ3\B with geometric boundary conditions on δM ≃ S2. A partial xistence result is obtained, giving a partial resolution of a conjecture of Bartnik on such static vacuum extensions. The existence and uniqueness of such extensions is closely related to Bartnik's definition of quasi-local mass.