Consider a triple of
“Bartnik data” (Σ,γ,H), where Σ is a topological 2-sphere with Riemannian
metric γ and positive function H. We view Bartnik data as a boundary
condition for the problem of finding a compact Riemannian 3-manifold (Ω,g) of
nonnegative scalar curvature whose boundary is isometric to (Σ,γ) with mean
curvature H. Considering the perturbed data (Σ,γ,λH) for a positive real
parameter λ, we find that such a “fill-in” (Ω,g) must exist for λ small and cannot
exist for λ 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 a nonnegativity property and is bounded
above by the Brown–York mass. However, our definition differs from many others 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.
Keywords
general relativity, quasi-local mass, scalar curvature,
static vacuum
