Positive mass theorem for initial data sets with arbitrary ends

Abstract: We showed a positive energy theorem for asymptotically flat initial data sets with the concept of spectral PSC by He-Shi-Yu, Bi-Hao-He-Shi-Zhu and Brendle-Wang. Then, we proved a quantitative shielding theorem concerning the causal property of the energy-momentum vector of an asymptotically hyperbolic manifold. As a result, we established the positive mass theorem for complete asymptotically hyperbolic manifolds satisfying the dominant energy condition. As corollaries, we also obtained corresponding results for manifolds with asymptotically locally hyperbolic ends with a certain symmetry.

• The paper establishes that complete asymptotically hyperbolic manifolds with scalar curvature at least -n(n1) have future-directed causal or zero energy-momentum in dimensions n 4 and above, without a spin assumption.
• The paper combines shielding estimates, Maskit-type gluing, hyperbolic patching, the Jang equation, and spectral positive scalar curvature to prove the positive energy theorem for asymptotically flat initial data in every dimension n 3 and above.
• The paper derives quantitative boundary and inextendibility results, showing how violations of causal mass or scalar-curvature hypotheses force incompleteness or detectable geometric defects near an end.

Overview

This paper by Tin-Yau Tsang establishes the positive mass theorem for complete asymptotically hyperbolic (AH) manifolds of arbitrary dimension $n \ge 4$ without any spin assumption, and, along the way, a positive energy theorem for asymptotically flat (AF) initial data sets in all dimensions $n \ge 3$. The main result states that if $(M^n,g)$ is complete with scalar curvature $R_g \ge -n(n-1)$ and possesses at least one AH end $\mathcal{E}$ of Hölder type $(\alpha,\delta)$ with $\alpha \in (0,1)$ and $\delta > n/2$, then the energy-momentum vector $m(\mathcal{E},g)$ is future-directed causal or vanishing. This extends to the non-spin setting the spin-based results of Chai–Wan [(2604.26978)'s cited literature], and parallels the AF arbitrary-ends program initiated by Lesourd–Unger–Yau.

The proof strategy synthesizes several recent techniques: Maskit-type gluing constructions of Chruściel–Delay, patching with hyperbolic space following Andersson–Cai–Galloway, the Jang equation reduction, and — crucially — the recent "spectral positive scalar curvature" framework developed by He–Shi–Yu, Bi–Hao–He–Shi–Zhu, and Brendle–Wang, which permits dimension induction via torical symmetrisation and thereby removes the dimensional restriction $n \le 7$ inherited from minimal hypersurface methods.

Quantitative shielding theorems

The technical core consists of two quantitative shielding statements. The first concerns AH manifolds: given nested neighborhoods $n \ge 3$0 of an end $n \ge 3$1 with distances $n \ge 3$2 and $n \ge 3$3, if $n \ge 3$4 has no points of incompleteness in $n \ge 3$5, satisfies $n \ge 3$6 there, and obeys the largeness condition

$n \ge 3$7

then the energy-momentum vector of $n \ge 3$8 is future-directed causal or vanishing. The second is the spacetime analogue for AF initial data sets $n \ge 3$9: under $(M^n,g)$0, $(M^n,g)$1 away from the end, and the largeness assumption $(M^n,g)$2 on the annular region, the ADM energy $(M^n,g)$3 is non-negative. These generalize Theorem 1.1 of Lee–Lesourd–Unger to the spacetime setting and to all dimensions. A boundary version (Theorem 1.7 analogue) controls the mean curvature of a compact inner boundary: for $(M^n,g)$4, the condition

$(M^n,g)$5

suffices; letting $(M^n,g)$6 recovers the classical bound $(M^n,g)$7, while $(M^n,g)$8 makes the shielding theorem applicable without any boundary mean curvature hypothesis at all.

Reduction to spectral positive scalar curvature

The proof of the AF positive energy theorem proceeds by modifying the second fundamental form. Given a smooth function $(M^n,g)$9, the tensor $R_g \ge -n(n-1)$0 transforms the constraint quantities as

$R_g \ge -n(n-1)$1

Because $R_g \ge -n(n-1)$2 away from the end, one can choose $R_g \ge -n(n-1)$3 so that the dominant energy condition holds on a region $R_g \ge -n(n-1)$4 containing the Euclidean end, while $R_g \ge -n(n-1)$5 near $R_g \ge -n(n-1)$6 forces every nearby homologous surface to be strictly outer trapped ($R_g \ge -n(n-1)$7). After imposing harmonic asymptotics and strict dominant energy condition via perturbation, the regularized Jang equation with blow-up yields a graph whose scalar curvature is positive in the strong spectral sense of He–Shi–Yu. Applying the Brendle–Wang dimension descent scheme (Theorem 1.4 of their preprint), valid in all dimensions thanks to torical symmetrisation and singularity removal, gives $R_g \ge -n(n-1)$8. For single-ended data, the perturbation machinery of Eichmair–Huang–Lee–Schoen applies directly. As a remark, the full inequality $R_g \ge -n(n-1)$9 follows from these positive energy theorems, and the author notes that Brendle–Wang independently obtained a comparable result via the regularized Jang equation, as did Hirsch–Khuri–Lesourd–Zhang.

From shielding to completeness

The passage from the quantitative shielding statement to the main theorem proceeds through a contradiction argument modeled on Chruściel–Delay and Chruściel–Galloway–Nguyen–Paetz. Assuming the energy-momentum vector is neither future-causal nor zero, the author first uses the "local" exotic gluing theorem of Chruściel–Delay to modify the metric near the conformal boundary, then doubles and glues two copies along half-hyperbolic spaces, producing a past-pointing timelike vector. A further perturbation renders the mass aspect function constant and negative, after which the Andersson–Cai–Galloway deformation makes the end exactly hyperbolic. This hyperbolic cap embeds isometrically into Minkowski space as a hyperboloid, allowing extension to a constant-time slice; the resulting AF initial data set satisfies the dominant energy condition with $\mathcal{E}$0. Feeding this configuration into the shielding argument produces a Jang graph component that is complete, has zero ADM energy, yet carries strictly positive spectral scalar curvature somewhere — contradicting the rigidity case of the spectral positive mass theorem.

For the complete-manifold theorem, completeness guarantees that the largeness parameter can always be met by choosing $\mathcal{E}$1 sufficiently large relative to the uniform lower bound $\mathcal{E}$2 on the modified energy density over the compact-in-the-core region, where $\mathcal{E}$3 is the Andersson–Cai–Galloway parameter.

Density theorem and inextendibility

An appendix proves a density theorem: any AH end with $\mathcal{E}$4 can be approximated in weighted Hölder norms by metrics with Wang's asymptotics, preserving the scalar curvature lower bound exactly outside a bounded set and controlling the mass functional to within $\mathcal{E}$5. The key technical departure from Lee–Lesourd–Unger is the use of the equation

$\mathcal{E}$6

whose constant barriers $\mathcal{E}$7 exist precisely because $\mathcal{E}$8; crucially, the barrier threshold $\mathcal{E}$9 depends only on asymptotic data, not on the exhaustion parameter, yielding global $(\alpha,\delta)$0 control and subsequential convergence.

As a corollary of the shielding theorem, the paper obtains an inextendibility statement: if the energy-momentum vector of an AH end fails to be future-causal or vanishing, then within a $(\alpha,\delta)$1-neighborhood of the end — with $(\alpha,\delta)$2 depending only on the mass and the weighted Hölder distance to hyperbolic space — either the scalar curvature drops below $(\alpha,\delta)$3 somewhere or an incomplete point exists. This quantifies precisely how violations of the hypotheses must manifest geometrically.

Asymptotically locally hyperbolic ends

Passing to universal covers in the manner of Chruściel–Galloway, the shielding theorem, the completeness theorem, the boundary mean curvature estimate, and the inextendibility corollary all carry over to asymptotically locally hyperbolic ends with topology $(\alpha,\delta)$4, where $(\alpha,\delta)$5 is a finite subgroup of $(\alpha,\delta)$6, with the scalar mass replacing the energy-momentum vector.

Limitations and open questions

Several restrictions are explicit. The AH results require $(\alpha,\delta)$7; the low-dimensional cases are covered by earlier work but the present gluing arguments are stated only above dimension three. The ALH extension requires the finite-cyclic-type symmetry $(\alpha,\delta)$8, leaving general ALH ends open. The density theorem assumes the integrability condition $(\alpha,\delta)$9 and Hölder-type decay with $\alpha \in (0,1)$0; whether the results extend under weaker asymptotics is not addressed. The appendix on unbounded singular sets of Jang graphs relies on Minkowski dimension bounds from Naber–Valtorta applied to almost-minimizing currents, and notes that the argument requires uniformly bounded $\alpha \in (0,1)$1 and $\alpha \in (0,1)$2 — a hypothesis satisfied here by strict dominance of the energy condition but not in full generality. Finally, the rigidity statement — characterizing when equality holds, i.e., when the manifold is hyperbolic space — is invoked from prior work rather than reproved, and its interaction with the incomplete-end setting remains tied to the assumptions of the cited spectral frameworks.

Conclusion

This paper completes, in the non-spin category, the causal characterization of the energy-momentum vector for complete manifolds with AH ends satisfying $\alpha \in (0,1)$3, in all dimensions $\alpha \in (0,1)$4, together with the corresponding positive energy theorem for AF initial data sets in all dimensions $\alpha \in (0,1)$5. The combination of Maskit gluing, hyperbolic patching, conformal density theory adapted to preserve scalar curvature lower bounds, and the spectral positive scalar curvature descent scheme provides a template likely applicable to further geometric inequalities on manifolds with controlled singularities or incompleteness.