In all dimensions , we introduce a differential operator of order 4 which, on Einstein manifolds, transforms the (fields of) trace free symmetric two tensors into TT-tensors. A large class of TT-tensors are obtained in this way, the restriction of our operator to these tensors being the composition of two shifted Lichnerowicz Laplacians.
En toutes dimensions , on introduit un opérateur différentiel d’ordre 4 qui, sur les variétés d’Einstein, transforme les (champs de) deux tenseurs symétriques de trace nulle en tenseurs TT. Une large classe de tenseurs TT est obtenue ainsi, la restriction de notre opérateur à ces tenseurs étant la composée de deux Laplaciens de Lichnerowicz translatés.
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Erwann Delay 1, 2
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@article{CRMATH_2023__361_G2_495_0,
author = {Erwann Delay},
title = {Une machine \`a tenseurs {TT} sur les vari\'et\'es {d{\textquoteright}Einstein}},
journal = {Comptes Rendus. Math\'ematique},
pages = {495--506},
publisher = {Acad\'emie des sciences, Paris},
volume = {361},
year = {2023},
doi = {10.5802/crmath.404},
language = {fr},
}
Erwann Delay. Une machine à tenseurs TT sur les variétés d’Einstein. Comptes Rendus. Mathématique, Volume 361 (2023), pp. 495-506. doi : 10.5802/crmath.404. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.404/
[1] TT-tensors and conformally flat structures on 3-manifolds, Mathematics of gravitation. Part I : Lorentzian geometry and Einstein equations (Warsaw, 1996) (Banach Center Publications), Volume 41, Polish Academy of Sciences, 1996, pp. 109-118
[2] On linearised vacuum constraint equations on Einstein manifolds, Class. Quant. Grav., Volume 37 (2020) no. 21, 14, p. 215012 | DOI | Zbl
[3] Some decompositions of the space of symmetric tensors on a Riemannian manifold, J. Differ. Equations, Volume 3 (1969), pp. 379-392
[4] Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, 10, Springer, 1987 | DOI | Numdam | Zbl
[5] Sur le noyau des opérateurs pseudo-différentiels à symbole surjectif et non injectif, C. R. Math. Acad. Sci. Paris, Volume 282 (1976), pp. 867-870 | Zbl
[6] On the existence and stability of the Penrose compactification, Ann. Henri Poincaré, Volume 8 (2007) no. 3, pp. 597-620 | DOI | Zbl
[7] On the stability of Riemannian manifold with parallel spinors, Invent. Math., Volume 161 (2005) no. 1, pp. 151-176 | Zbl
[8] Asymptotically Flat initial data with prescribed regularity at infinity, Commun. Math. Phys., Volume 222 (2001) no. 3, pp. 569-609 | DOI | Zbl
[9] Smooth compactly supported solutions of some underdetermined elliptic PDE, with gluing applications, Commun. Partial Differ. Equations, Volume 37 (2012) no. 10, pp. 1689-1716 | DOI | Zbl
[10] Inversion d’opérateurs de courbures au voisinage d’une métrique Ricci parallèle, Ann. Inst. Fourier, Volume 67 (2017) no. 2, pp. 521-538 | DOI | Zbl
[11] Linearization stability of the Einstein constraint equations on an asymptotically hyperbolic manifold, J. Math. Phys., Volume 51 (2010) no. 7, 072501, 14 pages | Zbl
[12] Rigidity and stability of Einstein metrics for quadratic curvature functionals, J. Reine Angew. Math., Volume 700 (2015), pp. 37-91 | Zbl
[13] Non-deformability of Einstein metrics, Osaka J. Math., Volume 15 (1978), pp. 419-433 | Zbl
[14] Variational Stability and Rigidity of Compact Einstein Manifolds, Quantum mathematical physics. A bridge between mathematics and physics (Regensburg, 2014), Birkhäuser/Springer, 2016, pp. 497-513 | Zbl
[15] Propagateurs et commutateurs en relativité générale, Publ. Math., Inst. Hautes Étud. Sci., Volume 10 (1961), pp. 293-344 | Numdam | Zbl
[16] Certain conditions for a Riemannian manifold to be isometric with a sphere, J. Math. Soc. Japan, Volume 14 (1962) no. 3, pp. 333-340 | Zbl
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