The positive mass theorem and distance estimates in the spin setting

Simone Cecchini, Rudolf Zeidler

Published 2021-08-26Version 1

Let $\mathcal{E} \subset M$ be an asymptotically Euclidean end in an otherwise arbitrary complete and connected Riemannian manifold $(M,g)$ of non-negative scalar curvature. We use an augmentation of Witten's proof of the positive mass theorem to show that the ADM-mass of $(\mathcal{E},g)$ must be non-negative. This answers Schoen and Yau's question on the positive mass theorem with arbitrary ends in the case of spin manifolds. Without the spin condition this result has recently been obtained in dimensions $\leq 7$ by Lesourd, Unger and Yau under Schwarzschild asymptotics on the end $\mathcal{E}$. In addition, we are able prove a rigidity theorem in our setting, that is, if the mass of $\mathcal{E}$ is zero, then $(M,g)$ must be flat. We also obtain 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.