  We prove the existence and local uniqueness of asymptotically flat, static
vacuum metrics with arbitrarily prescribed Bartnik boundary data that are close
to the induced boundary data on any star-shaped hypersurface or a general
family of perturbed hypersurfaces in the Euclidean space. It confirms the
existence part of the Bartnik static extension conjecture for large classes of
boundary data, and the static vacuum metric obtained is geometrically unique in
a neighborhood of the Euclidean metric.
