The Positive Mass Theorem with Arbitrary Ends

Abstract

We prove a Riemannian positive mass theorem for manifolds with a single asymptotically flat end, but otherwise arbitrary other ends, which can be incomplete and contain negative scalar curvature. The incompleteness and negativity is compensated for by large positive scalar curvature on an annulus, in a quantitative fashion. In the complete noncompact case with nonnegative scalar curvature, we have no extra assumption and hence prove a long-standing conjecture of Schoen and Yau. Contents 1. Introduction 1 2. The Dirichlet problem for µ-bubbles 6 3. Construction of the weight 9 4. Proof of Theorem 1.6 with an extra assumption 11 4.1. Construction of Σ ∞ 11 4.2. Asymptotics of Σ ∞ 14 4.3. Strong stability of Σ ∞ 23 4.4. Finishing the argument when R g > 0 far out 27 5. Proofs of the main theorems 28 Appendix A. Curvature estimate in homogeneously regular manifolds 30 References 34

The positive mass theorem and distance estimates in the spin setting

“…The value of the number R(E, g) > 0 appearing in Theorem A could in principle be traced through our proof but it crucially depends on the constant in a weighted Poincaré inequality on the chosen asymptotically Euclidean end and thus it appears difficult to make explicit. Note that Lesourd, Unger, and Yau have also established a quantitative theorem in their approach to Conjecture 1.2 which involves more explicit estimates; see [18,Theorem 1.6]. However, unlike our Theorem A, their quantitative theorem relies on a largeness assumption on the scalar curvature which forces it to be strictly positive in a certain region.…”

“…However, the last remaining cases of the Liouville theorem have recently been proved through a combination of preprints by Chodosh and Li [7] and Lesourd, Unger, and Yau [17], but without addressing Conjecture 1.2. Instead, in a separate more recent preprint, Lesourd, Unger, and Yau [18] proved Conjecture 1.2 for n ≤ 7 assuming that the chosen end E satisfies a more specific fall-off condition, namely that it is asymptotic to Schwarzschild (compare Example 2.2). In the classical case of Theorem 1.1, more general fall-off conditions can be reduced to Schwarzschild asymptotics via a density theorem, but at the moment this does not appear to be readily available in the setting of arbitrary ends as pointed out in [18].…”

“…Instead, in a separate more recent preprint, Lesourd, Unger, and Yau [18] proved Conjecture 1.2 for n ≤ 7 assuming that the chosen end E satisfies a more specific fall-off condition, namely that it is asymptotic to Schwarzschild (compare Example 2.2). In the classical case of Theorem 1.1, more general fall-off conditions can be reduced to Schwarzschild asymptotics via a density theorem, but at the moment this does not appear to be readily available in the setting of arbitrary ends as pointed out in [18]. Thus, so far, Conjecture 1.2 still remains open under the general fall-off conditions now commonly used in the context of the positive mass theorem as well as in higher dimensions.…”

“…However, unlike our Theorem A, their quantitative theorem relies on a largeness assumption on the scalar curvature which forces it to be strictly positive in a certain region. Therefore in [18] an intermediary step involving a conformal perturbation of the metric is necessary in order to attack Conjecture 1.2. Since in the situation of Conjecture 1.2 the part outside of the chosen end is an arbitrary complete manifold, this is a subtle point and here the stronger Schwarzschild asymptotics assumed in [18] play an important role.…”

“…Therefore in [18] an intermediary step involving a conformal perturbation of the metric is necessary in order to attack Conjecture 1.2. Since in the situation of Conjecture 1.2 the part outside of the chosen end is an arbitrary complete manifold, this is a subtle point and here the stronger Schwarzschild asymptotics assumed in [18] play an important role. In comparison, our approach completely circumvents this issue by leaving the metric in place.…”