  In the context of the Bartnik mass, there are two fundamentally different
notions of an extension of some compact Riemannian manifold $(\Omega,\gamma)$
with boundary. In one case, the extension is taken to be a manifold without
boundary in which $(\Omega,\gamma)$ embeds isometrically, and in the other case
the extension is taken to be a manifold with boundary where the boundary data
is determined by $\partial \Omega$.
  We give a type of convexity condition under which we can say both of these
types of extensions indeed yield the same value for the Bartnik mass. Under the
same hypotheses we prove that the Bartnik mass varies continuously with respect
to the boundary data. This also provides a method to use estimates for the
Bartnik mass of constant mean curvature (CMC) Bartnik data, to obtain estimates
for the Bartnik mass of non-CMC Bartnik data. The key idea for these results is
a method for gluing Bartnik extensions of given Bartnik data to other nearby
Bartnik data.
