The positive mass theorem and distance estimates in the spin setting (Q6571602)

The positive mass theorem of \textit{R. Schoen} and \textit{S.-T. Yau} [Proc. Natl. Acad. Sci. USA 76, 1024--1025 (1979; Zbl 0415.53029)] is a central result in the study of the geometry of scalar curvature, mathematical relativity and geometric analysis. It says that a complete asymptotically Euclidean manifold (without boundary) of dimension \(n\geq 3\) which has non-negative scalar curvature has non-negative ADM-mass in each end. Furthermore, if an end has zero mass, then the manifold is isometric to Euclidean space.\N\NA curious feature of this is that it consists of separate statements for each asymptotically Euclidean end of the manifold. Thus one is lead to ask if the positive mass theorem can, in a certain sense, be localized to a single end without imposing the strong topological and metric restrictions of being asymptotically Euclidean on the remaining part of the manifold.\N\NConjecture. Let \((M, g)\) be an arbitrary complete Riemannian manifold of dimension \(n\geq 3\) which has non-negative scalar curvature. Let \(E \subset M\) be a single asymptotically Euclidean end in \(M\). Then \(E\) has non-negative mass.\N\NThe main goal of the paper is to study the above conjecture in the spin setting. The authors show that, if the mass of an asymptotically Euclidean end \(E\) in a spin manifold \(M\) is negative, then the hypotheses of the positive mass theorem must be violated in an \(R\)-neighborhood around \(E\), where \(R\) is a constant depending only on \(E\) and not on the entire manifold. In particular, they deduce that the conjecture holds for spin manifolds, thus recovering a result by \textit{R. A. Bartnik} and \textit{P. T. Chruściel} [J. Reine Angew. Math. 579, 13--73 (2005; Zbl 1174.58305)].