  Consider a triple of "Bartnik data" $(\Sigma, \gamma,H)$, where $\Sigma$ is a
topological 2-sphere with Riemannian metric $\gamma$ and positive function $H$.
We view Bartnik data as a boundary condition for the problem of finding a
compact Riemannian 3-manifold $(\Omega,g)$ of nonnegative scalar curvature
whose boundary is isometric to $(\Sigma,\gamma)$ with mean curvature $H$.
Considering the perturbed data $(\Sigma, \gamma, \lambda H)$ for a positive
real parameter $\lambda$, we find that such a "fill-in" $(\Omega,g)$ must exist
for $\lambda$ small and cannot exist for $\lambda$ large; moreover, we prove
there exists an intermediate threshold value.
  The main application is the construction of a new quasi-local mass, a concept
of interest in general relativity. This mass has the nonnegativity property,
but differs from many other definitions in that it tends to vanish on static
vacuum (as opposed to flat) regions. We also recognize this mass as a special
case of a type of twisted product of quasi-local mass functionals. Several
ideas in this paper draw on work of Bray, Brendle--Marques--Neves, Corvino,
Miao, and Shi--Tam.
