标题： 正质量定理和自旋情形下的距离估计 显示英文标题

标题： The positive mass theorem and distance estimates in the spin setting

摘要： 设$\mathcal{E}$是一个渐近欧几里得端，在其余部分为任意的完备且连通的黎曼自旋流形$(M,g)$中。 我们证明，如果$\mathcal{E}$具有负的 ADM-质量，则存在一个仅依赖于$\mathcal{E}$的常数$R > 0$，使得$M$在$M$中围绕$\mathcal{E}$的$R$邻域内必须不完整或具有负标量曲率的点。 这为 Schoen 和 Yau 关于自旋流形上任意端点的正质量定理的问题提供了定量回答。 最近，Lesourd、Unger 和 Yau 在维度$\leq 7$中未使用自旋条件的情况下，假设端部具有 Schwarzschild 渐进行为，得到了类似的结果$\mathcal{E}$。 我们还推导出在所选端部的某些区域内标量曲率均匀正时的显式定量距离估计$\mathcal{E}$。 在这里，我们得到的改进常数类似于 Gromov 的带有标量曲率的度量不等式。 显示英文摘要

摘要： Let $\mathcal{E}$ be an asymptotically Euclidean end in an otherwise arbitrary complete and connected Riemannian spin manifold $(M,g)$. We show that if $\mathcal{E}$ has negative ADM-mass, then there exists a constant $R > 0$, depending only on $\mathcal{E}$, such that $M$ must become incomplete or have a point of negative scalar curvature in the $R$-neighborhood around $\mathcal{E}$ in $M$. This gives a quantitative answer to Schoen and Yau's question on the positive mass theorem with arbitrary ends for spin manifolds. Similar results have recently been obtained by Lesourd, Unger and Yau without the spin condition in dimensions $\leq 7$ assuming Schwarzschild asymptotics on the end $\mathcal{E}$. We also derive explicit quantitative distance estimates in case the scalar curvature is uniformly positive in some region of the chosen end $\mathcal{E}$. Here we obtain refined constants reminiscent of Gromov's metric inequalities with scalar curvature.