Positive Mass Theorems for Spin Initial Data Sets With Arbitrary Ends and Dominant Energy Shields

Abstract

Abstract 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.