The Pohozaev-Schoen identity on asymptotically Euclidean manifolds: Conservation laws and their applications

Avalos, R.; Freitas, A.

doi:10.1016/j.anihpc.2021.01.002

Avalos, R.¹ ; Freitas, A.²

¹ a Departamento de Matemática, Universidade Federal do Ceará, R. Humberto Monte, 60455-760, Fortaleza, CE, Brazil
² b Departamento de Matemática, Universidade Federal da Paraíba, 58059-900, João Pessoa, Paraíba, Brazil

Annales de l'I.H.P. Analyse non linéaire, novembre – décembre 2021, Tome 38 (2021) no. 6, pp. 1703-1724.

Le texte intégral des articles récents est réservé aux abonnés de la revue. Consultez l'article sur le site de la revue.

Résumé

The aim of this paper is to present a version of the generalized Pohozaev-Schoen identity in the context of asymptotically Euclidean manifolds. Since these kind of geometric identities have proven to be a very powerful tool when analysing different geometric problems for compact manifolds, we will present a variety of applications within this new context. Among these applications, we will show some rigidity results for asymptotically Euclidean Ricci-solitons and Codazzi-solitons. Also, we will present an almost-Schur type inequality valid in this non-compact setting which does not need restrictions on the Ricci curvature. Finally, we will show how some rigidity results related with static potentials also follow from these type of conservation principles.

MR Zbl

DOI : 10.1016/j.anihpc.2021.01.002

Classification : 53C21, 53C24
Mots-clés : Pohozaev-Schoen identity, Asymptotically Euclidean manifolds, Generalized solitons, Almost-Schur lemma, Static metrics

Affiliations des auteurs :

Avalos, R. ¹ ; Freitas, A. ²

@article{AIHPC_2021__38_6_1703_0,
     author = {Avalos, R. and Freitas, A.},
     title = {The {Pohozaev-Schoen} identity on asymptotically {Euclidean} manifolds: {Conservation} laws and their applications},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {1703--1724},
     publisher = {Elsevier},
     volume = {38},
     number = {6},
     year = {2021},
     doi = {10.1016/j.anihpc.2021.01.002},
     mrnumber = {4327894},
     zbl = {1483.53058},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1016/j.anihpc.2021.01.002/}
}

TY  - JOUR
AU  - Avalos, R.
AU  - Freitas, A.
TI  - The Pohozaev-Schoen identity on asymptotically Euclidean manifolds: Conservation laws and their applications
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2021
SP  - 1703
EP  - 1724
VL  - 38
IS  - 6
PB  - Elsevier
UR  - http://archive.numdam.org/articles/10.1016/j.anihpc.2021.01.002/
DO  - 10.1016/j.anihpc.2021.01.002
LA  - en
ID  - AIHPC_2021__38_6_1703_0
ER  -

%0 Journal Article
%A Avalos, R.
%A Freitas, A.
%T The Pohozaev-Schoen identity on asymptotically Euclidean manifolds: Conservation laws and their applications
%J Annales de l'I.H.P. Analyse non linéaire
%D 2021
%P 1703-1724
%V 38
%N 6
%I Elsevier
%U http://archive.numdam.org/articles/10.1016/j.anihpc.2021.01.002/
%R 10.1016/j.anihpc.2021.01.002
%G en
%F AIHPC_2021__38_6_1703_0

Avalos, R.; Freitas, A. The Pohozaev-Schoen identity on asymptotically Euclidean manifolds: Conservation laws and their applications. Annales de l'I.H.P. Analyse non linéaire, novembre – décembre 2021, Tome 38 (2021) no. 6, pp. 1703-1724. doi : 10.1016/j.anihpc.2021.01.002. http://archive.numdam.org/articles/10.1016/j.anihpc.2021.01.002/

Bibliographie
Cité par

[1] Alías, L.J.; de Lira, J.H.; Rigoli, M. Mean curvature flow solitons in the presence of conformal vector fields | arXiv | DOI | Zbl

[2] Barbosa, E. A note on the almost-Schur lemma on 4-dimensional Riemannian closed manifolds, Proc. Am. Math. Soc., Volume 140 (2012) no. 12, pp. 4319-4322 | DOI | MR | Zbl

[3] Barbosa, E.; Freitas, A.; Lima, L. The generalized Pohozaev-Schoen identity and some geometric applications, Commun. Anal. Geom., Volume 28 (2020) no. 2, pp. 223-242 | DOI | MR | Zbl

[4] Bartnik, R. The mass of an asymptotically flat manifold, Commun. Pure Appl. Math., Volume 39 (1986) | DOI | MR | Zbl

[5] Cheng, X. An almost-Schur type lemma for symmetric (2, 0) tensors and applications, Pac. J. Math., Volume 267 (2014), p. 2 | DOI | MR | Zbl

[6] Cheng, X.; Zhou, D. Rigidity for closed totally umbilical hypersurfaces in space forms, J. Geom. Anal., Volume 24 (2014), pp. 1337-1345 | DOI | MR | Zbl

[7] Choquet-Bruhat, Y. General Relativity and the Einstein Equations, Oxford University Press Inc., New York, 2009 | MR | Zbl

[8] Choquet-Bruhat, Y.; Christodoulou, D. Elliptic systems in $H_{s, δ}$ spaces on manifolds which are Euclidean at infinity, Acta Math., Volume 145 (1981), pp. 129-150 | DOI | MR | Zbl

[9] Christodoulou, D.; O'Murchada, N. The boost problem in general relativity, Commun. Math. Phys., Volume 80 (1981), pp. 271-300 | DOI | MR | Zbl

[10] Chrusciel, P.T.; Mazzeo, R. On ‘many-black-hole’ vacuum spacetimes, Class. Quantum Gravity, Volume 20 (2003), p. 729 | DOI | MR | Zbl

[11] Corvino, J. Scalar curvature deformation and a gluing construction for the Einstein constraint equations, Commun. Math. Phys., Volume 214 (2000), pp. 137-189 | DOI | MR | Zbl

[12] de Lellis, C.; Topping, P.M. Almost Schur lemma, Calc. Var., Volume 43 (2012), pp. 347-354 | DOI | MR | Zbl

[13] Dilts, J.; Maxwell, D. Yamabe classification and prescribed scalar curvature in the asymptotically Euclidean setting, Commun. Anal. Geom., Volume 26 (2018) no. 5, pp. 1127-1168 | DOI | MR | Zbl

[14] Eichmair, M.; Galloway, G.J.; Pollack, D. Topological censorship from the initial data point of view, J. Differ. Geom., Volume 95 (2013), pp. 389-405 | DOI | MR | Zbl

[15] Gover, A.R.; Orsted, B. Universal principles for Kazdan-Warner and Pohozaev-Schoen type identities, Commun. Contemp. Math., Volume 15 (2013) no. 4 | DOI | MR | Zbl

[16] Hang, F.; Yang, P.C. Lectures on the fourth-order Q curvature equation, Geometric Analysis Around Scalar Curvatures, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., vol. 31, World Sci. Publ., Hackensack, NJ, 2016, pp. 1-33 | DOI | MR | Zbl

[17] Hawking, S.W. Black holes in general relativity, Commun. Math. Phys., Volume 25 (1972), pp. 152-166 | DOI | MR

[18] Huisken, G.; Ilmanen, Tom The inverse mean curvature flow and the Riemannian Pensrose inequality, J. Differ. Geom., Volume 59 (2001), pp. 353-437 | DOI | MR | Zbl

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

[20] Li, Y. Ricci flow on asymptotically Euclidean manifolds, Geom. Topol., Volume 22 (2018), pp. 1837-1891 | DOI | MR | Zbl

[21] Lin, Y.J.; Yuan, W. A symmetric $2$ -tensor canonically associated to $Q$ -curvature and its applications, Pac. J. Math., Volume 291 (2017), pp. 425-438 | DOI | MR | Zbl

[22] Maxwell, D. Solutions of the Einstein constraint equations with apparent horizon boundaries, Commun. Math. Phys., Volume 253 (2004), pp. 561-583 | DOI | MR | Zbl

[23] Miao, P.; Tam, L.F. Static potentials on asymptotically flat manifolds, Ann. Henri Poincaré, Volume 16 (2015), p. 2239 | DOI | MR | Zbl

[24] Miao, P.; Tam, L.F. Evaluation of the ADM mass and center of mass via the Ricci tensor, Proc. Am. Math. Soc., Volume 144 (2016), pp. 753-761 | DOI | MR | Zbl

[25] Pigola, S.; Rigoli, M.; Rimoldi, M.; Setti, A.G. Ricci almost solitons, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5), Volume X (2011), pp. 757-799 | Numdam | MR | Zbl

[26] Schoen, R. The existence of weak solutions with prescribed singular behaviour for a conformally invariant scalar equation, Commun. Pure Appl. Math., Volume 41 (1988) no. 3, pp. 317-392 | DOI | MR | Zbl

[27] Schoen, R.; Yau, S.T. On the proof of the positive mass conjecture in general relativity, Commun. Math. Phys., Volume 65 (1979), pp. 45-76 | DOI | MR | Zbl

[28] Schoen, R.; Yau, S.T. Positive scalar curvature and minimal hypersurface singularities | arXiv | DOI | Zbl

[29] Taylor, M.E. Partial Differential Equations I, Springer-Verlag New York Inc., New York, 1996 | MR | Zbl

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

Cité par Sources :