Noncompact Fill-Ins of Bartnik Data

Abstract

We generalize Y. Shi and L.-F.\ Tam's \cite{ShiTam} nonnegativity result for the Brown-York mass, by considering nonnegative scalar curvature (NNSC) fill-ins that need only be complete rather than compact. Moreover, the NNSC fill-ins need not even be complete as long the incompleteness is ``shielded'' by a region with positive scalar curvature and occurs occurs sufficiently far away. We accomplish this by generalizing P.~Miao's~\cite{Miao02} positive mass theorem with corners to asymptotically flat manifolds that may have other complete ends, or possibly incomplete ends that are appropriately shielded. We can similarly extend other results on the compact NNSC fill-in problem to allow for complete (or shielded) NNSC fill-ins. In particular, we prove the following generalization of a theorem of Miao~\cite{Miao20}: Given any metric $γ$ on a closed manifold $Σ^{n-1}$, there exists a constant $λ$ such that for any complete (or shielded) NNSC fill-in $(Ω^n, g)$ of $(Σ^{n-1},γ)$, we have $\min_ΣH \le λ$, where $H$ is the mean curvature of $Σ$ with respect to $g$.