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Fill-ins of nonnegative scalar curvature, static metrics, and quasi-local mass. (English) Zbl 1268.53043

Summary: 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 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.

MSC:

53C20 Global Riemannian geometry, including pinching
83C99 General relativity
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