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The spacetime positive mass theorem in dimensions less than eight
Open accessAbstract
We prove the spacetime positive mass theorem in dimensions less than eight. This theorem asserts that for any asymptotically flat initial data set that satisfies the dominant energy condition, the inequality E ≥ | P | holds, where (E, P) is the ADM energy-momentum vector. Previously, this theorem was only known for spin manifolds [38]. Our approach is a modification of the minimal hypersurface technique that was used by the last named author and S.-T. Yau to establish the time-symmetric case of this theorem [30, 27]. Instead of minimal hypersurfaces, we use marginally outer trapped hypersurfaces (MOTS) whose existence is guaranteed by earlier work of the first named author [14]. An important part of our proof is to introduce an appropriate substitute for the area functional that is used in the time-symmetric case to single out certain minimal hyper surfaces. We also establish a density theorem of independent interest and use it to reduce the general case of the spacetime positive mass theorem to the special case of initial data that has harmonic asymptotics and satisfies the strict dominant energy condition.
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TL;DR
The paper proves the spacetime positive mass theorem \(E \ge |P|\) for asymptotically flat initial data satisfying the dominant energy condition in dimensions \(3 \le n < 8\), without a spin assumption. The proof is by contradiction and induction, using marginally outer trapped hypersurfaces and a density theorem reducing to harmonic asymptotics with strict dominant energy condition.
Key findings
- Main theorem: for \(3 \le n < 8\), any \(n\)-dimensional asymptotically flat initial data set \((M,g,k)\) satisfying the dominant energy condition has ADM energy-momentum vector \((E,P)\) satisfying \(E \ge |P|\).
- The proof is self-contained rather than by reduction to a known case; it gives a new proof of the \(E \ge 0\) theorem for non-time-symmetric data.
- The \(n=3\) case is completed by contradiction to the Gauss-Bonnet theorem; for \(3 < n < 8\), the proof is by induction and contradicts the time-symmetric positive mass theorem in dimension \(n-1\).
- A density theorem shows that an asymptotically flat initial data set satisfying the dominant energy condition can be perturbed to one with harmonic asymptotics and strict dominant energy condition \(\mu > |J|\), with small changes in \(g\), \(\pi\), \(E\), and \(P\).
Methods
- Proof by contradiction and induction on dimension. Assuming \(E < |P|\), the authors use the density theorem to assume harmonic asymptotics and strict dominant energy condition, with \(P\) pointing in the vertical direction.
- They construct stable marginally outer trapped hypersurfaces (MOTS) \(\Sigma_{\rho,h}\) in vertical cylinders \(C_\rho\) between barrier planes, then extract a complete MOTS limit \(\Sigma_\infty\) as \(\rho \to \infty\).
- For \(n=3\), stability and Gauss-Bonnet give a contradiction. For \(3 < n < 8\), they introduce a functional \(F\), pick a height \(h_\rho\) and vector field \(X\) satisfying \(D\theta^+|_{\Sigma_{\rho,h_\rho}}(X)=0\) and \(DF|_{\Sigma_{\rho,h_\rho}}(X)\ge 0\), derive a stability-like inequality, and use a conformal change to produce an asymptotically flat scalar-flat metric of negative energy, contradicting the time-symmetric theorem in dimension \(n-1\).
- The theory of \(\lambda\)-minimizing currents substitutes for a variational principle for MOTS; the assumption \(n<8\) ensures the relevant MOTS are smooth.
Limitations
- The paper states that the theorem does not include a characterization of the equality case \(E=|P|\); the natural conjecture is that \(E=|P|=0\) and \((M,g)\) embeds isometrically into Minkowski space with second fundamental form \(k\), but the proof by contradiction means the equality case requires a substantially new idea.
- The paper states that the restriction \(n<8\) is used only for regularity of the MOTS; for \(n>8\), the lemma still holds except that the \(\lambda\)-minimizing boundaries are only regular away from a thin singular set.
- Evident from the text: the method is non-constructive for the equality case and relies on perturbation to harmonic asymptotics and strict dominant energy condition, so it does not directly address rigidity or higher-dimensional singularities.
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“Proposition 4 (Strong maximum principle).Let g be a C 2 Riemannian metric on M = Bn-1 1 (0) × (-2, 2) ⊂ R n .Let F be a C 1 function on the unit sphere bundle of M and let u 1 , u 2 ∈ C 2 ( Bn-1 1 (0)) be such that -1 ≤ u 1 (x ) ≤ u 2 (x ) ≤ 1 for all x ∈ Bn-1 1 (0).”
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“If in addition (f, V ) ∈ C 0,α -n-q 0 for some q 0 > 0 and α ∈ (0, 1 -n/p], and (g, π ) ∈ C 2,α loc × C 1,α loc , then (v, Z) ∈ C 2,α 2-n .”
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“Lemma 6.Let (M, g, k) and be as described above.For every sufficiently large ρ and all h ∈ [-, ] there exists a stable C 3,α MOTS ρ,h in C ρ, whose boundary equals ρ,h and which meets ∂C ρ transversely.”
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“There exists δ > 0 depending only on , n, p, q, q 0 , (g -g E , π ) W 2,p -q ×W 1,p -q-1 , and (µ, J ) -( μ, J ) L 1 -n-q 0 /2 such that if g -ḡ W 2,p -q ≤ δ and π -π W 1,p -1-q ≤ δ, then |E -Ē| < and |P -P | < .”
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“In the last section, we show that an initial data set which satisfies the dominant energy condition can be perturbed by a small amount to one with harmonic asymptotics that satisfies the strict dominant energy condition.”
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“Together with Eichmair and Schoen, the authors proved that the positive mass inequality E ≥ |P | holds in dimensions less than eight [11] by using marginally outer trapped hypersurfaces (MOTS) in place of the minimal hypersurfaces used in the Schoen-Yau proof of the Riemannian positive mass theorem.”
From Equality in the Spacetime Positive Mass Theorem (2020) · cites this paper as [11]
In the citing paper
“The modified constraint map Φ (g,π) : M 2,p -q ×W 1,p -1-q -→ L p -2-q is smooth, and DΦ (g,π) : W 2,p -q × W 1,p -1-q -→ L p -2-q is surjective.”
From Equality in the Spacetime Positive Mass Theorem (2020) · cites this paper as [11]
In the citing paper
“There exists λ 0 > 0, C 1 > 0 such that for each 0 < λ < λ 0 , there exists an initial data set (ḡ, π) of the same type with (ḡ, π) -(g, π) W 2,p -q ×W 1,p -1-q < C 1 λ such that μ > (1 + λ)| J | ḡ + (1 + λ)(µ -|J| g ) where (μ, J) are the mass and current densities of (ḡ, π).As a consequence, if µ ≥ |J| g , then (ḡ, π) satisfies the strict dominant energy condition with μ -| J | ḡ > λ| J | ḡ.”
From Equality in the Spacetime Positive Mass Theorem (2020) · cites this paper as [11]
In the citing paper
“if an asymptotically flat initial data set satisfies the dominant energy condition and has E = |P |, then E = |P | = 0”
From Equality in the Spacetime Positive Mass Theorem (2020) · cites this paper as [11]
In the citing paper
“As mentioned above in the introduction, the positive mass inequality for asymptotically flat manifolds of type (p, q, q 0 , α) was proved in [11] for 3 ≤ n ≤ 7 and has been announced in [15] for n ≥ 3.”
From Equality in the Spacetime Positive Mass Theorem (2020) · cites this paper as [11]
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Tilted spacetime positive mass theorem with arbitrary ends
Liu, D. (2024) · Journal of Geometry and Physics · doi:10.1016/j.geomphys.2024.105238
A survey on the positive mass theorem for asymptotically flat initial data sets
Huang, L.-H. (2025) · Comptes Rendus Mécanique · doi:10.5802/crmeca.270
Self-citationJaracz, J. S. (2025) · International Journal of Geometric Methods in Modern Physics · doi:10.1142/s0219887825501981
Meco, B. (2025) · Annals of Global Analysis and Geometry · doi:10.1007/s10455-025-10013-4
Metadata
- Authors
- Eichmair, M., Huang, L.-H., Lee, D. A., & Schoen, R.
- Affiliations
- University of Vienna (AT)
- University of Connecticut (US)
- City University of New York (US)
- University of California, Irvine (US)
- Venue
- Journal of the European Mathematical Society
- Published
- 2015-12-16
- Volume
- 18
- Issue
- 1
- Pages
- 83-121
- ISSN
- 1435-9855 (ISSN-L)
- 1435-9863
- OpenAlex
- W1703994914
- Semantic Scholar
- 2c16d1ac6608f440856f4e3df2ca3c89c39fc106
- Cited by
- 82
- References
- 36
- Citations by year
- 2026: 2 · 2025: 8 · 2024: 12 · 2023: 3 · 2022: 10 · 2021: 13 · 2020: 3 · 2019: 4 · 2018: 4 · 2017: 4 · 2016: 5 · 2015: 8
- FWCI
- 6.75
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Locations
- Journal of the European Mathematical Society (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- arXiv (Cornell University)
- Funders
- National Science Foundation (1404966, 1005560, 0903467, 1105323, 0906038, 1308837)
- Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung (140467, 200021)
- Topics
- Physical Sciences › Mathematics › Applied Mathematics › Geometric Analysis and Curvature Flows
- Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Relativity and Gravitational Theory
- Physical Sciences › Physics and Astronomy › Nuclear and High Energy Physics › Black Holes and Theoretical Physics
- Keywords
- asymptotically flat initial data, dominant energy condition, minimal hypersurfaces, density theorem
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
A language model drew the claims from the paper on 2026-11-14, under version 3 of the claim definition (model deepseek-v4.1-flash, prompt extract-v3).
A second model labelled each claim on 2026-11-14 (model jev-1.13-20260917, definition revision gold-guideline-v0.2).
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
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Data sources
Bibliographic data: Semantic Scholar, OpenAlex, Crossref.
Contains information from the Semantic Scholar Academic Graph Datasets, made available under the ODC Attribution License (ODC-BY 1.0).
Semantic Scholar’s data is described in: Kinney, R., et al. (2023). The Semantic Scholar Open Data Platform. arXiv:2301.10140.
