  Given a Riemannian 3-ball $(\bar B, g)$ of non-negative scalar curvature,
Bartnik conjectured that $(\bar B, g)$ admits an asymptotically flat (AF)
extension (without horizons) of the least possible ADM mass, and that such a
mass-minimizer is an AF solution to the static vacuum Einstein equations,
uniquely determined by natural geometric conditions on the boundary data of
$(\bar B, g)$.
  We prove the validity of the second statement, i.e.~such mass-minimizers, if
they exist, are indeed AF solutions of the static vacuum equations. On the
other hand, we prove that the first statement is not true in general; there is
a rather large class of bodies $(\bar B, g)$ for which a minimal mass extension
does not exist.
