Abstract
Motivated by the quasi-local mass problem in general relativity, we apply the asymptotically flat extensions, constructed by Shi and Tam in the proof of the positivity of the Brown-York mass, to study a fill-in problem of realizing geometric data on a 2-sphere as the boundary of a compact 3-manifold of non-negative scalar curvature. We characterize the relationship between two borderline cases: one in which the Shi-Tam extension has zero total mass, and another in which fill-ins of non-negative scalar curvature fail to exist. Additionally, we prove a type of positive mass theorem in the latter case.
