The positive mass theorem and distance estimates in the spin setting

Abstract

Let E ⊂ 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 (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 ≤ 7 by Lesourd, Unger and Yau under Schwarzschild asymptotics on the end E. In addition, we are able prove a rigidity theorem in our setting, that is, if the mass of 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 E. Here we obtain refined constants reminiscent of Gromov's metric inequalities with scalar curvature.

A note on the generalized Geroch conjecture for complete spin manifolds

“…If 3 ≤ dim M ≤ 7, then Conjecture 0.3 has been proved for arbitrary M by Chodosh and Li [2], with the case of dim M = 3 also proved by Lesourd-Unger-Yau [5]. A recent paper by Zhu [17] shows that Conjecture 0.3 implies the positive mass theorem with arbitrary ends, which in the spin setting has been proved in Cecchini-Zeidler [1,Theorem B]. Thus Theorem 0.2 gives an alternate proof of the positive mass theorem with arbitrary ends in the spin setting.…”

Singular positive mass theorem with arbitrary ends

“…The shielding principle for Riemannian manifolds with nonnegative scalar curvature yields that boundary far behind a domain with positive scalar curvature behaves like mean-convex. Such philosophy has appeared in many works, for instances see [4,18]. In this work, we consider the Georoch conjecture using the same perspective.…”

Density and positive mass theorems for incomplete manifolds

“…Using Dirac operators on a space with a strong weight on the other ends, R. Bartnik and P. Chruściel are able to prove a remarkable spacetime positive mass theorem with arbitary ends under a spin assumption [BC05]. After [LUY21] appeared, S. Cecchini and R. Zeidler revisted the Riemannian positive mass theorem in the spin setting [CZ21]. They use Callias operators (i.e.…”