The positive mass theorem and distance estimates in the spin setting

Abstract

Let E \mathcal {E} be an asymptotically Euclidean end in an otherwise arbitrary connected Riemannian spin manifold ( M , g ) (M,g) . We show that if E \mathcal {E} has negative ADM-mass, then there exists a constant R > 0 R > 0 , depending only on E \mathcal {E} , such that M M must become incomplete or have a point of negative scalar curvature in the R R -neighborhood around E \mathcal {E} in M 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 [Positive scalar curvature on noncompact manifolds and the liouville theorem, 2020; The positive mass theorem with arbitrary ends, 2021] without the spin condition in dimensions ≤ 7 \leq 7 assuming Schwarzschild asymptotics on the end E \mathcal {E} . We also derive explicit quantitative distance estimates in case the scalar curvature is uniformly positive in some region of the chosen end E \mathcal {E} . Here we obtain refined constants reminiscent of Gromov’s metric inequalities with scalar curvature.

Nonnegative scalar curvature on manifolds with at least two ends

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