Bartnik数据的非紧填充
Noncompact Fill-Ins of Bartnik Data
摘要：
我们推广Y。Shi和L.- F. Tam的\cite{ShiTam}非负布朗约克质量的结果，通过考虑非负标量曲率（NNSC）填充，只需要是完整的，而不是紧凑的。此外，中立国监委会的填补甚至不需要是完整的，只要不完整性被具有正标量曲率的区域“屏蔽”，并且发生在足够远的地方。我们通过推广P来实现这一点。Miao的~ \cite{Miao02}正质量定理与角到渐近平坦流形，可能有其他完整的结束，或可能不完整的结束，适当屏蔽。我们可以类似地扩展关于紧NNSC填充问题的其他结果，以允许完全特别地，我们证明了以下Miao~ \cite{Miao20}定理的推广：给定闭流形$Σ^{n-1}$上的任何度量$γ$，存在常数$λ$，使得对于任何完备的（或屏蔽）NNSC填充$(Σ^{n-1},γ)$的$(Ω^n, g)$，我们有$\min_ΣH \le λ$，其中$H$是$Σ$相对于$g$的平均曲率。
英文摘要：
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 $g$.
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