  In [Comm. Anal. Geom., 13(5):845-885, 2005.], Bartnik described the phase
space for the Einstein equations, modelled on weighted Sobolev spaces with
local regularity $(g,\pi)\in H^2\times H^1$. In particular, it was established
that the space of solutions to the contraints form a Hilbert submanifold of
this phase space. The motivation for this work was to study the quasilocal mass
functional now bearing his name. However, the phase space considered there was
over a manifold without boundary. Here we demonstrate that analogous results
hold in the case where the manifold has an interior compact boundary, where
Dirichlet boundary conditions are imposed on the metric. Then, still following
Bartnik's work, we demonstrate the critical points of the mass functional over
this space of extensions correspond to stationary solutions. Furthermore, if
this solution is sufficiently regular then it is in fact a static black hole
solution. In particular, in the vacuum case, critical points only occur at
exterior Schwarzschild solutions; that is, critical points of the mass over
this space do not exist generically. Finally, we briefly discuss the case when
the boundary data is Bartnik's geometric data.
