Compactness of solutions to the Yamabe problem

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. To cite this article: Y.Y. Li, L. Zhang, C. R. Acad. Sci. Paris, Ser. I 338 (2004).  2004 Académie des sciences. Published by Elsevier SAS. All rights reserved. Résumé Compacité des solutions du problème de Yamabe. 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. Pour citer cet article : Y.Y. Li, L. Zhang, C. R. Acad. Sci. Paris, Ser. I 338 (2004).  2004 Académie des sciences. Published by Elsevier SAS. All rights reserved. Version française abrégée Soit (M,g) une variété riemannienne régulière, compacte et connexe sans bord de dimension n. On considère l’équation de Yamabe − gu+ n− 2 4(n− 1)Rgu= u (n+2)/(n−2), u > 0, sur M, (1) où g désigne l’opérateur de Laplace–Beltrami. Soit M= {u | u ∈ C2(M), u vérifie (1)}. On considère les deux cas suivant : E-mail addresses: [email protected] (Y.Y. Li), [email protected] (L. Zhang). 1631-073X/$ – see front matter  2004 Académie des sciences. Published by Elsevier SAS. All rights reserved. doi:10.1016/j.crma.2004.02.018 694 Y.Y. Li, L. Zhang / C. R. Acad. Sci. Paris, Ser. I 338 (2004) 693–695 1◦. Dimension de M 7. 2◦. Dimension de M 8 et |Wg |> 0 sur M , où Wg désigne le tenseur de Weyl de g. Théorème 0.1. On suppose que (M,g) n’est pas conformément équivalent à la sphère standard. On fait l’hypothèse 1◦ ou 2◦. Alors il existe une constante C dependant seulement de (M,g) telle que ‖u‖L∞(M) C ∀u ∈M. 1. The Yamabe conjecture Let (M,g) be an n-dimensional smooth compact Riemannian manifold without boundary. For n 3, the Yamabe conjecture states that there exist metrics which are pointwise conformal to g and have constant scalar curvature. The Yamabe conjecture was proved through the works of Yamabe [11], Trudinger [10], Aubin [1] and Schoen [8]. Different proofs in the case n 5 and in the case (M,g) is locally conformally flat were given by Bahri and Brezis [3] and Bahri [2]. Consider the Yamabe equation − gu+ n− 2 4(n− 1)Rgu= u (n+2)/(n−2), u > 0, on M, (2) where g denotes the Laplace–Beltrami operator. Let M= {u | u ∈ C2(M), u satisfies (2)}. For n 3, under the assumption that (M,g) is locally conformally flat and is not conformally diffeomorphic to standard spheres, Schoen proved in 1991, see [9], that for any non-negative integer k, ‖u‖Ck(M,g) C, ∀u ∈M, (3) where C is some constant depending only on (M,g) and k. He also announced in the same paper the same result for general Riemannian manifolds, without the locally conformally flatness assumption. The proof of this claim has not been made available. For general Riemannian manifolds of dimension n= 3, a proof was given by Li and Zhu in [7]; while for dimension n= 4, the combination of results of Li and Zhang [5] and Druet [4] yields a proof.