Partial Differential Equations
Compactness of solutions to the Yamabe problem
Comptes Rendus. Mathématique, Volume 338 (2004) no. 9, pp. 693-695.

We establish compactness of solutions to the Yamabe problem on any smooth compact connected Riemannian manifold (not conformally diffeomorphic to standard spheres) of dimension n⩽7 as well as on any manifold of dimension n⩾8 under some additional hypothesis.

On établit la compacité des solutions du problème de Yamabe sur toute variété riemannienne, régulière compacte connexe (non conformément équivalente à la sphère standard) de dimension n⩽7. Le même résultat est valable en dimension n⩾8 sous une hypothèse supplémentaire.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2004.02.018
Li, YanYan 1; Zhang, Lei 2

1 Department of Mathematics, Rutgers University, 110 Frelinghuysen Rd., Piscataway, NJ 08854, USA
2 Department of Mathematics, Texas A&M University, 3368 TAMU, College Station, TX 77843-3368, USA
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Li, YanYan; Zhang, Lei. Compactness of solutions to the Yamabe problem. Comptes Rendus. Mathématique, Volume 338 (2004) no. 9, pp. 693-695. doi : 10.1016/j.crma.2004.02.018. http://www.numdam.org/articles/10.1016/j.crma.2004.02.018/

[1] Aubin, T. Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire, J. Math. Pures Appl., Volume 55 (1976), pp. 269-296

[2] Bahri, A. Another proof of the Yamabe conjecture for locally conformally flat manifolds, Nonlinear Anal., Volume 20 (1993), pp. 1261-1278

[3] Bahri, A.; Brezis, H. Non-linear elliptic equations on Riemannian manifolds with the Sobolev critical exponent, Topics in Geometry, Progr. Nonlinear Differential Equations Appl., vol. 20, Birkhäuser Boston, Boston, MA, 1996, pp. 1-100

[4] O. Druet, From one bubble to several bubbles. The low-dimensional case, J. Differential Geometry, in press

[5] Y.Y. Li, L. Zhang, A Harnack type inequality for the Yamabe equation in low dimensions, Calc. Var. Partial Differential Equations, in press

[6] Y.Y. Li, L. Zhang, Compactness of solutions to the Yamabe problem, in preparation

[7] Li, Y.Y.; Zhu, M. Yamabe type equations on three-dimensional Riemannian manifolds, Commun. Contemp. Math., Volume 1 (1999), pp. 1-50

[8] Schoen, R. Conformal deformation of a Riemannian metric to constant scalar curvature, J. Differential Geom., Volume 20 (1984), pp. 479-495

[9] Schoen, R. On the number of constant scalar curvature metrics in a conformal class (Lawson, H.B.; Tenenblat, K., eds.), Differential Geometry: A Symposium in Honor of Manfredo Do Carmo, Wiley, 1991, pp. 311-320

[10] Trudinger, N. Remarks concerning the conformal deformation of Riemannian structures on compact manifolds, Ann. Scuola Norm. Sup. Cl. Sci., Volume 22 (1968) no. 3, pp. 265-274

[11] Yamabe, H. On a deformation of Riemannian structures on compact manifolds, Osaka Math. J., Volume 12 (1960), pp. 21-37

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