[Enquête sur le théorème de la masse positive pour les ensembles de données initiales asymptotiquement plats]
En l’honneur du 100e anniversaire d’Yvonne Choquet-Bruhat, nous présentons cette étude sur le théorème de la masse positive. Issu d’une conjecture en relativité générale concernant la masse de l’ADM, le théorème de la masse positive a influencé de manière significative la géométrie et l’analyse au cours des quatre dernières décennies et continue d’inspirer de nouvelles connexions. Nous passons en revue les contributions fondamentales ainsi que les avancées récentes, puis nous concentrons nos discussions sur le cas de l’égalité et les contre-exemples provenant des espaces-temps d’ondes pp.
In honor of Yvonne Choquet-Bruhat’s 100th birthday, we present this survey on the positive mass theorem. Originating from a conjecture in general relativity regarding the ADM mass, the positive mass theorem has significantly influenced geometry and analysis over the past four decades and continued to inspire new connections. We review seminal contributions as well as recent advances, and then we focus our discussions on the equality case and the counter-examples arising from pp-wave spacetimes.
Accepté le :
Publié le :
Mots-clés : Masse ADM, Énergie-momentum ADM, Ensemble de données initiales asymptotiquement plates, Condition d’énergie dominante, Onde pp
Lan-Hsuan Huang 1
CC-BY 4.0
@article{CRMECA_2025__353_G1_177_0,
author = {Lan-Hsuan Huang},
title = {A survey on the positive mass theorem for asymptotically flat initial data sets},
journal = {Comptes Rendus. M\'ecanique},
pages = {177--194},
publisher = {Acad\'emie des sciences, Paris},
volume = {353},
year = {2025},
doi = {10.5802/crmeca.270},
language = {en},
}
Lan-Hsuan Huang. A survey on the positive mass theorem for asymptotically flat initial data sets. Comptes Rendus. Mécanique, Volume 353 (2025), pp. 177-194. doi : 10.5802/crmeca.270. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.5802/crmeca.270/
[1] Beginnings of the Cauchy problem for Einstein’s field equations, Surveys in Differential Geometry 2015. One Hundred Years of General Relativity (Surveys in Differential Geometry), Volume 20, International Press, Boston, MA, 2015, pp. 1-16 | DOI
[2] General relativity and the Einstein equations, Oxford Mathematical Monographs, Oxford University Press, Oxford, 2009, p. xxvi+785
[3] Solution of the local mass problem in general relativity, Commun. Math. Phys., Volume 51 (1976) no. 3, pp. 283-296 http://projecteuclid.org/euclid.cmp/1103900391 | DOI
[4] Variational methods and positive energy in general relativity, Ann. Phys., Volume 50 (1968) no. 3, pp. 548-570 | DOI
[5] General relativity, Differential Geometry (Proceedings of Symposia in Pure Mathematics, Vol. XXVII, Part 2, Stanford University, Stanford, California, 1973), American Mathematical Society, Providence, RI, 1975, pp. 401-414
[6] Conformal deformation of a Riemannian metric to constant scalar curvature, J. Differ. Geom., Volume 20 (1984) no. 2, pp. 479-495 http://projecteuclid.org/getRecord?id=euclid.jdg/1214439291 | DOI
[7] On the proof of the positive mass conjecture in general relativity, Commun. Math. Phys., Volume 65 (1979) no. 1, pp. 45-76 http://projecteuclid.org/getRecord?id=euclid.cmp/1103904790 | DOI
[8] The energy and the linear momentum of space-times in general relativity, Commun. Math. Phys., Volume 79 (1981) no. 1, pp. 47-51 http://projecteuclid.org/getRecord?id=euclid.cmp/1103908887 | DOI
[9] Complete manifolds with nonnegative scalar curvature and the positive action conjecture in general relativity, Proc. Nat. Acad. Sci. USA, Volume 76 (1979) no. 3, pp. 1024-1025 | DOI
[10] Variational theory for the total scalar curvature functional for Riemannian metrics and related topics, Topics in Calculus of Variations (Montecatini Terme, 1987) (Lecture Notes in Mathematics), Volume 1365, Springer, 1989, pp. 120-154 | DOI
[11] Scalar curvature and hammocks, Math. Ann., Volume 313 (1999) no. 3, pp. 385-407 | DOI
[12] Existence of incompressible minimal surfaces and the topology of three-dimensional manifolds with nonnegative scalar curvature, Ann. Math. (2), Volume 110 (1979) no. 1, pp. 127-142 | DOI
[13] The inverse mean curvature flow and the Riemannian Penrose inequality, J. Differ. Geom., Volume 59 (2001) no. 3, pp. 353-437 http://projecteuclid.org/getRecord?id=euclid.jdg/1090349447 | DOI
[14] Ricci flow on asymptotically Euclidean manifolds, Geom. Topol., Volume 22 (2018) no. 3, pp. 1837-1891 | DOI
[15] Harmonic functions and the mass of 3-dimensional asymptotically flat Riemannian manifolds, J. Geom. Anal., Volume 32 (2022) no. 6, 184 | DOI
[16] A Green’s function proof of the positive mass theorem, Commun. Math. Phys., Volume 405 (2024) no. 2, 54 | DOI
[17] Proof of the positive mass theorem. II, Commun. Math. Phys., Volume 79 (1981) no. 2, pp. 231-260 | DOI
[18] The Jang equation reduction of the spacetime positive energy theorem in dimensions less than eight, Commun. Math. Phys., Volume 319 (2013) no. 3, pp. 575-593 | DOI
[19] A new proof of the positive energy theorem, Commun. Math. Phys., Volume 80 (1981) no. 3, pp. 381-402 | DOI
[20] On Witten’s proof of the positive energy theorem, Commun. Math. Phys., Volume 84 (1982) no. 2, pp. 223-238 http://projecteuclid.org/getRecord?id=euclid.cmp/1103921154 | DOI
[21] The mass of an asymptotically flat manifold, Commun. Pure Appl. Math., Volume 39 (1986) no. 5, pp. 661-693 | DOI
[22] Positive mass theorems for higher dimensional Lorentzian manifolds, J. Math. Phys., Volume 49 (2008) no. 2, 022504 | DOI
[23] The spacetime positive mass theorem in dimensions less than eight, J. Eur. Math. Soc. (JEMS), Volume 18 (2016) no. 1, pp. 83-121 | DOI
[24] Spacetime harmonic functions and the mass of 3-dimensional asymptotically flat initial data for the Einstein equations, J. Differ. Geom., Volume 122 (2022) no. 2, pp. 223-258 | DOI
[25] Skin Structures on Minimal Hypersurfaces, Preprint, 2015 | arXiv
[26] Hyperbolic Geometry and Potential Theory on Minimal Hypersurfaces, Preprint, 2015 | arXiv
[27] Skin Structures in Scalar Curvature Geometry, Preprint, 2015 ([math.DG]) | arXiv
[28] The Higher Dimensional Positive Mass Theorem II, Preprint, 2016 ([math.DG]) | arXiv
[29] Positive scalar curvature and minimal hypersurface singularities, Surveys in Differential Geometry 2019. Differential Geometry, Calabi-Yau Theory, and General Relativity. Part 2 (Surveys in Differential Geometry), Volume 24, Internatial Press, Boston, MA, 2022, pp. 441-480
[30] Generic regularity of homologically area minimizing hypersurfaces in eight-dimensional manifolds, Commun. Anal. Geom., Volume 1 (1993) no. 2, pp. 217-228 | DOI
[31] Generic regularity for minimizing hypersurfaces in dimensions 9 and 10, Preprint, 2023 ([math.DG]) | arXiv
[32] A positive mass theorem for two spatial dimensions, Preprint ([gr-qc]) | arXiv
[33] Bartnik mass minimizing initial data sets and improvability of the dominant energy scalar, J. Differ. Geom., Volume 126 (2024) no. 2, pp. 741-800 | DOI
[34] Energy-momentum of isolated systems cannot be null, Phys. Lett. A, Volume 89 (1982) no. 4, pp. 181-184 | DOI
[35] A strictly-positive mass theorem, Commun. Math. Phys., Volume 108 (1987) no. 4, pp. 653-665 http://projecteuclid.org/euclid.cmp/1104116629 | DOI
[36] Killing vectors in asymptotically flat space-times. I. Asymptotically translational Killing vectors and the rigid positive energy theorem, J. Math. Phys., Volume 37 (1996) no. 4, pp. 1939-1961 | DOI
[37] Killing vectors in asymptotically flat space-times. II. Asymptotically translational Killing vectors and the rigid positive energy theorem in higher dimensions, J. Math. Phys., Volume 47 (2006) no. 2, 022502 | DOI
[38] Phase space for the Einstein equations, Commun. Anal. Geom., Volume 13 (2005) no. 5, pp. 845-885 http://projecteuclid.org/euclid.cag/1144438298 | DOI
[39] Equality in the spacetime positive mass theorem II, Preprint, 2023 ([math.DG]) | arXiv
[40] The case of equality for the spacetime positive mass theorem, J. Geom. Anal., Volume 33 (2023) no. 1, 30 | DOI
[41] Initial data sets with vanishing mass are contained in pp-wave spacetimes, Preprint, 2024 ([gr-qc]) | arXiv
[42] The mathematics of Richard Schoen, Not. Amer. Math. Soc., Volume 65 (2018) no. 11, pp. 1349-1376
[43] Trapped surfaces, topology of black holes, and the positive mass theorem, Not. Amer. Math. Soc., Volume 69 (2022) no. 4, pp. 536-545 | DOI
[44] Geometric Relativity, Graduate Studies in Mathematics, 201, American Mathematical Society, Providence, RI, 2019 (xii+361 pages.) | DOI
[45] The mass of asymptotically hyperbolic manifolds, J. Differ. Geom., Volume 57 (2001) no. 2, pp. 273-299 http://projecteuclid.org/euclid.jdg/1090348112 | DOI
[46] The mass of asymptotically hyperbolic Riemannian manifolds, Pacific J. Math., Volume 212 (2003) no. 2, pp. 231-264 | DOI
[47] Rigidity and positivity of mass for asymptotically hyperbolic manifolds, Ann. Henri Poincaré, Volume 9 (2008) no. 1, pp. 1-33 | DOI
[48] On the mass aspect function and positive energy theorems for asymptotically hyperbolic manifolds, Class. Quantum Grav., Volume 35 (2018), 115015
[49] Mass rigidity for hyperbolic manifolds, Commun. Math. Phys., Volume 376 (2020) no. 3, pp. 2329-2349 | DOI
[50] The Jang equation and the positive mass theorem in the asymptotically hyperbolic setting, Commun. Math. Phys., Volume 386 (2021) no. 2, pp. 903-973 | DOI
[51] Localized deformation for initial data sets with the dominant energy condition, Calc. Var. Partial Differ. Equations, Volume 59 (2020) no. 1, 42 | DOI
[52] On the invariant mass conjecture in general relativity, Commun. Math. Phys., Volume 120 (1988) no. 2, pp. 233-248 http://projecteuclid.org/euclid.cmp/1104177748 | DOI
[53] Role of surface integrals in the Hamiltonian formulation of general relativity, Ann. Phys., Volume 88 (1974), pp. 286-318 | DOI
[54] Spacetime symmetries and linearization stability of the Einstein equations. I, J. Math. Phys., Volume 16 (1975), pp. 493-498 | DOI
[55] The size of isoperimetric surfaces in 3-manifolds and a rigidity result for the upper hemisphere, Proc. Amer. Math. Soc., Volume 137 (2009) no. 8, pp. 2733-2740 | DOI
[56] The boost problem in general relativity, Commun. Math. Phys., Volume 80 (1981) no. 2, pp. 271-300 http://projecteuclid.org/euclid.cmp/1103919879 | DOI
[57] On the energy of the gravitational field at spatial infinity, Conference on Mathematical Relativity (Canberra, 1988) (Proceedings of the Centre for Mathematical Analysis, Australian National University), Volume 19, 1989
[58] Spacetime harmonic functions and applications to mass, Perspectives in Scalar Curvature, Volume 2, World Scientific Publishing, Hackensack, NJ, 2023, pp. 593-639 | DOI
[59] Equality in the spacetime positive mass theorem, Commun. Math. Phys., Volume 376 (2020) no. 3, pp. 2379-2407 | DOI
Cité par Sources :
Commentaires - Politique