Principe de recollement des équations des contraintes en relativité générale
[Gluing principle for the constraint equations in general relativity]
Julien Cortier1
1 Institut des Hautes Études Scientifiques, 35, route de Chartres, 91440 Bures-sur-Yvette, France
Séminaire de théorie spectrale et géométrie, Volume 30 (2011-2012), pp. 21-45.

The “gluing” method to find solutions to the relativistic constraint equations is reviewed. In particular, we describe the Corvino-Schoen method to construct families of solutions on a non-compact manifold with prescribed geometry on an asymptotic end, with emphasis on the “non-localized” gluing. We then provide a list of results obtained by various authors using such techniques, including the question of gluing Riemannian metrics while preserving their (constant) scalar curvature. We eventually give some applications in geometric analysis and in general relativity.

La méthode de «  recollement  » permettant de trouver des solutions des équations des contraintes relativistes est décrite. En particulier, on expose la méthode de Corvino-Schoen pour construire des familles de solutions sur une variété non-compacte avec géométrie prescrite sur un bout asymptotique, en insistant sur le recollement «  non-localisé  ». Une liste de résultats obtenus par divers auteurs à partir de telles techniques est alors fournie, incluant la question du recollement de métriques riemanniennes en préservant leur courbure scalaire (constante). On donne enfin certaines applications en analyse géométrique et en relativité générale.

DOI: 10.5802/tsg.289
Classification: 58J05, 83C05, 53C21, 53C50
Mot clés : relativité générale, formulation de Cauchy, équations des contraintes, recollement.
Keywords: general relativity, Cauchy formulation, constraint equations, gluing.
Julien Cortier 1

1 Institut des Hautes Études Scientifiques, 35, route de Chartres, 91440 Bures-sur-Yvette, France 
@article{TSG_2011-2012__30__21_0,
     author = {Julien Cortier},
     title = {Principe de recollement des \'equations des~contraintes en relativit\'e g\'en\'erale},
     journal = {S\'eminaire de th\'eorie spectrale et g\'eom\'etrie},
     pages = {21--45},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {30},
     year = {2011-2012},
     doi = {10.5802/tsg.289},
     language = {fr},
     url = {https://proceedings.centre-mersenne.org/articles/10.5802/tsg.289/}
}
TY  - JOUR
AU  - Julien Cortier
TI  - Principe de recollement des équations des contraintes en relativité générale
JO  - Séminaire de théorie spectrale et géométrie
PY  - 2011-2012
SP  - 21
EP  - 45
VL  - 30
PB  - Institut Fourier
PP  - Grenoble
UR  - https://proceedings.centre-mersenne.org/articles/10.5802/tsg.289/
DO  - 10.5802/tsg.289
LA  - fr
ID  - TSG_2011-2012__30__21_0
ER  - 
%0 Journal Article
%A Julien Cortier
%T Principe de recollement des équations des contraintes en relativité générale
%J Séminaire de théorie spectrale et géométrie
%D 2011-2012
%P 21-45
%V 30
%I Institut Fourier
%C Grenoble
%U https://proceedings.centre-mersenne.org/articles/10.5802/tsg.289/
%R 10.5802/tsg.289
%G fr
%F TSG_2011-2012__30__21_0
Julien Cortier. Principe de recollement des équations des contraintes en relativité générale. Séminaire de théorie spectrale et géométrie, Volume 30 (2011-2012), pp. 21-45. doi : 10.5802/tsg.289. https://proceedings.centre-mersenne.org/articles/10.5802/tsg.289/

[1] M.T. Anderson; M.A. Khuri On the Bartnik extension problem for the static vacuum Einstein equations, Class. Quantum Grav., Volume 30 (2013) (125005) | MR | Zbl

[2] R. Arnowitt; S. Deser; C. Misner Coordinate invariance and energy expressions in general relativity, Phys. Rev., Volume 122 (1961), pp. 997-1006 (arXiv : gr-qc/0405109) | MR | Zbl

[3] R. Bartnik The mass of an asymptotically flat manifold, Commun. Pure Appl. Math., Volume 39 (1986), pp. 661-693 | MR | Zbl

[4] R. Bartnik; J. Isenberg The constraint equations, The Einstein equations and the large scale behavior of gravitational fields : 50 years of the Cauchy problem in general relativity (eds P.T. Chruściel, H. Friedrich), Birkhauser Basel, Switzerland, 2004, pp. 1-34 | MR | Zbl

[5] R. Beig; P.T. Chruściel Killing Initial Data, Class. Quantum Grav., Volume 14 (1997) no. 1A, p. A83-A92 (arXiv : gr-qc/9604040) | MR | Zbl

[6] R. Beig; P.T. Chruściel; R.M. Schoen KIDs are non-generic, Ann. H. Poincaré, Volume 6 (2005) no. 1, pp. 155-194 | MR | Zbl

[7] Y. Choquet-Bruhat; R. Geroch Global aspects of the Cauchy problem in general relativity, Comm. Math. Phys., Volume 14 (1969), pp. 329-335 | MR | Zbl

[8] P.T. Chruściel On the invariant mass conjecture in general relativity, Commun. Math. Phys., Volume 120 (1988), pp. 233-248 | MR | Zbl

[9] P.T. Chruściel; J. Corvino; J. Isenberg Construction of N-Body Initial Data Sets in General Relativity, Commun. Math. Phys., Volume 304 (2011) no. 3, pp. 637-647 | MR | Zbl

[10] P.T. Chruściel; E. Delay On mapping properties of the general relativistic constraints operator in weighted function spaces, with applications, Mém. Soc. Math. Fr. (2003) no. 94 | Numdam | MR | Zbl

[11] P.T. Chruściel; E. Delay Gluing constructions for asymptotically hyperbolic manifolds with constant scalar curvature, Comm. Anal. Geom., Volume 17 (2009) no. 2, pp. 343-381 | MR | Zbl

[12] P.T. Chruściel; M. Herzlich The mass of asymptotically hyperbolic Riemannian manifolds, Pacific J. Math, Volume 212 (2003) no. 2, pp. 231-264 | MR | Zbl

[13] P.T. Chruściel; J. Isenberg; D. Pollack Initial data engineering, Commun. Math. Phys., Volume 257 (2005) no. 1, pp. 29-42 | MR | Zbl

[14] P.T. Chruściel; J. Jezierski; S. Leski The Trautman-Bondi mass of hyperboloidal initial data sets, Adv. Theor. Math. Phys., Volume 8 (2004) no. 1, pp. 83-139 (arxiv.org/pdf/gr-qc/0307109.pdf) | MR | Zbl

[15] P.T. Chruściel; F. Pacard; D. Pollack Singular Yamabe metrics and initial data with exactly Kottler-Schwarzschild-de Sitter ends II, Math. Res. Lett., Volume 16 (2009) no. 1, pp. 157-164 | MR | Zbl

[16] P.T. Chruściel; D. Pollack Singular Yamabe metrics and initial data with exactly Kottler-Schwarzschild-de Sitter ends, Ann. H. Poincaré, Volume 9 (2008), pp. 639-654 | MR | Zbl

[17] J. Cortier A family of asymptotically hyperbolic manifolds with arbitrary energy-momentum vectors, J. Math. Phys., Volume 53 (2012) no. 10 http://link.aip.org/link/?JMP/53/102504 (arXiv : 1205.1377v2 [math.DG]) | DOI | MR

[18] J. Cortier Gluing construction of initial data with Kerr-de Sitter ends, Ann. H. Poincaré, Volume 14 (2013) no. 5, pp. 1109-1134 | DOI | MR | Zbl

[19] J. Corvino Scalar curvature deformation and a gluing construction for the Einstein constraint equations, Comm. Math. Phys., Volume 214 (2000) no. 1, pp. 137-189 | MR | Zbl

[20] J. Corvino; M. Eichmair; P. Miao Deformation of scalar curvature and volume, Math. Ann. (2013) (Online DOI 10.1007/s00208-013-0903-8) | MR

[21] J. Corvino; R.M. Schoen On the asymptotics for the vacuum Einstein constraint equations, J. Diff. Geom., Volume 73 (2006) no. 2, pp. 185-217 | MR | Zbl

[22] E. Delay Localized gluing of Riemannian metrics in interpolating their scalar curvature, Diff. Geom. Appl., Volume 29 (2011) no. 3, pp. 433-439 | MR | Zbl

[23] E. Delay Smooth compactly supported solutions of some underdetermined elliptic PDE, with gluing applications, Comm. PDE, Volume 37 (2012) no. 10, pp. 1689-1716 | MR | Zbl

[24] E. Delay; L. Mazzieri Refined gluing for vacuum Einstein constraint equations (2010) (arXiv : 1003.4178 [math.DG])

[25] Y. Fourès-Bruhat Théorèmes d’existence pour certains systèmes d’équations aux dérivées partielles non linéaires, Acta Math, Volume 88 (1952), pp. 141-225 | MR | Zbl

[26] M. Herzlich Théorèmes de masse positive, Séminaire de Théorie Spectrale et Géométrie, Volume 16 (1998), pp. 107-126 | Numdam | MR | Zbl

[27] E. Humbert Relativité générale (d’après M. Vaugon) et quelques problèmes mathématiques qui en sont issus (2010) (arXiv : 1004.2402v4 [math.DG])

[28] J. Isenberg; D. Maxwell; D. Pollack A gluing construction for non-vacuum solutions of the Einstein contraint equations, Adv. Theor. Math. Phys., Volume 9 (2005) no. 1, pp. 129-172 (arXiv : gr-qc/0501083) | MR | Zbl

[29] J.L. Jauregui Fill-ins of nonnegative scalar curvature, static metrics, and quasi-local mass, Pacific. J. Math., Volume 261 (2013) no. 2, pp. 417-444 | MR | Zbl

[30] J.L. Kazdan Positive energy in general relativity, Séminaire N. Bourbaki, Volume 120 (1982) no. 593, pp. 315-330 | Numdam | MR | Zbl

[31] J. M. Lee; T. H. Parker The Yamabe problem, Bull. Amer. Math. Soc. (N.S.), Volume 17 (1987) no. 1, pp. 37-91 | MR | Zbl

[32] H. Lindblad; I. Rodnianski Global Existence for the Einstein Vacuum Equations in Wave Coordinates, Comm. Math. Phys., Volume 256 (2005) no. 1, pp. 43-110 | MR | Zbl

[33] H. Lindblad; I. Rodnianski The global stability of Minkowski space-time in harmonic gauge, Ann. Math., Volume 171 (2010) no. 3, pp. 1401-1477 | MR | Zbl

[34] D. Maerten Killing initial data revisited, J. Math. Phys., Volume 45 (2004) no. 7, pp. 2594-2599 | DOI | MR | Zbl

[35] L. Mazzieri Generalized gluing for Einstein constraint equations, Calc. Var., Volume 34 (2009), pp. 453-473 | MR

[36] B. Michel Geometric invariance of mass-like asymptotic invariants, J. Math. Phys., Volume 52 (2011) no. 5, pp. 052504, 14 pp. (arXiv : 1012.3775v2 [math-ph]) | MR

[37] V. Moncrief Space-time symmetries and linearization stability of the Einstein equations. I, J. Math. Phys., Volume 16 (1975) no. 3, pp. 493-498 | MR | Zbl

[38] V. Moncrief Space-time symmetries and linearization stability of the Einstein equations. II, J. Math. Phys., Volume 17 (1976) no. 10, pp. 1893-1902 | MR

[39] R.M. Schoen 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 Berlin / Heidelberg, 1989, pp. 120-154 | MR | Zbl

[40] R.M. Schoen; S.T. Yau On the proof of the positive mass conjecture in general relativity, Commun. Math. Phys., Volume 65 (1979) no. 1, pp. 45-76 | MR | Zbl

[41] R.M. Schoen; S.T. Yau Proof of the positive mass theorem II, Commun. Math. Phys., Volume 79 (1981), pp. 231-260 | MR | Zbl

[42] E. Witten A simple proof of the positive energy theorem, Commun. Math. Phys., Volume 80 (1981), pp. 381-402 | MR | Zbl

Cited by Sources: