We prove a positive mass theorem for spin initial data sets $(M,g,k)$ that contain an asymptotically flat end and a shield of dominant energy (a subset of $M$ on which the dominant energy scalar $\mu -|J|$ has a positive lower bound). In a similar vein, we show that for an asymptotically flat end $\mathcal{E}$ that violates the positive mass theorem (i.e., $\textrm{E} < |\textrm{P}|$), there exists a constant $R>0$, depending only on $\mathcal{E}$, such that any initial data set containing $\mathcal{E}$ must violate the hypotheses of Witten’s proof of the positive mass theorem in an $R$-neighborhood of $\mathcal{E}$. This implies the positive mass theorem for spin initial data sets with arbitrary ends, and we also prove a rigidity statement. Our proofs are based on a modification of Witten’s approach to the positive mass theorem involving an additional independent timelike direction in the spinor bundle.

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具有任意末端和主能量屏蔽的自旋初始数据集的正质量定理

我们证明了自旋初始数据集 $(M,g,k)$ 的正质量定理，该数据集包含渐近平端和主导能量屏蔽（$M$ 的子集，其上主导能量标量 $\mu -| J|$ 具有正下限）。类似地，我们证明，对于违反正质量定理的渐近平端 $\mathcal{E}$ （即 $\textrm{E} < |\textrm{P}|$），存在常数$R>0$，仅取决于$\mathcal{E}$，因此任何包含$\mathcal{E}$的初始数据集都必须违反$R$-中的威滕正质量定理证明的假设$\mathcal{E}$ 的邻域。这意味着具有任意末端的自旋初始数据集的正质量定理，并且我们还证明了刚性陈述。我们的证明基于对威滕正质量定理方法的修改，涉及旋量丛中的附加独立类时方向。