On a fully nonlinear Yamabe problem

Ge, Yuxin; Wang, Guofang

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 39 (2006) no. 4, pp. 569-598.

[1] Andrews B., Monotone quantities and unique limits for evolving convex hypersurfaces, Internat. Math. Res. Notices 1997 (1997) 1001-1031. | MR | Zbl

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

[3] Aubin T., Li Y., On the best Sobolev inequality, J. Math. Pures Appl. 78 (1999) 353-387. | MR | Zbl

[4] Brendle S., Viaclovsky J., A variational characterization for ${\sigma }_{n/2}$, Calc. Var. Partial Differential Equations 20 (2004) 399-402. | MR | Zbl

[5] Caffarelli L., Nirenberg L., Spruck J., The Dirichlet problem for nonlinear second-order elliptic equations. III. Functions of the eigenvalues of the Hessian, Acta Math. 155 (1985) 261-301. | MR | Zbl

[6] Chang A., Gursky M., Yang P., An equation of Monge-Ampère type in conformal geometry, and four manifolds of positive Ricci curvature, Ann. of Math. 155 (2002) 709-787. | MR | Zbl

[7] Chang A., Gursky M., Yang P., An a priori estimate for a fully nonlinear equation on Four-manifolds, J. Anal. Math. 87 (2002) 151-186. | MR | Zbl

[8] Chang A., Gursky M., Yang P., Entire solutions of a fully nonlinear equation, in: Lectures on Partial Differential Equations, New Stud. Adv. Math., vol. 2, Int. Press, Somerville, MA, 2003, pp. 43-60. | MR | Zbl

[9] Chou K.-S., On a real Monge-Ampère functional (K.S. Tso), Invent. Math. 101 (1990) 425-448. | EuDML | MR | Zbl

[10] Chou K.-S., Wang X.-J., A variational theory of the Hessian equation, Comm. Pure Appl. Math. 54 (2001) 1029-1064. | MR | Zbl

[11] Garding L., An inequality for hyperbolic polynomials, J. Math. Mech. 8 (1959) 957-965. | MR | Zbl

[12] Ge Y., Wang G., On a conformal quotient equation, in preparation.

[13] González M. d. M., Removability of singularities for a class of fully non-linear elliptic equations Preprint, 2004. | MR

[14] Gromov M., Lawson H.B., The classification of simply connected manifolds of positive scalar curvature, Ann. of Math. (2) 111 (1980) 423-434. | MR | Zbl

[15] Guan P., Lin C.-S., Wang G., Schouten tensor and some topological properties, Comm. Anal. Geom. 13 (2005) 845-860. | MR | Zbl

[16] Guan P., Lin C.-S., Wang G., Local gradient estimates for conformal quotient equations, Preprint.

[17] Guan P., Viaclovsky J., Wang G., Some properties of the Schouten tensor and applications to conformal geometry, Trans. Amer. Math. Soc. 355 (2003) 925-933. | MR | Zbl

[18] Guan P., Wang G., Local estimates for a class of fully nonlinear equations arising from conformal geometry, Int. Math. Res. Not. 2003 (2003) 1413-1432. | MR | Zbl

[19] Guan P., Wang G., A fully nonlinear conformal flow on locally conformally flat manifolds, J. reine angew. Math. 557 (2003) 219-238. | MR | Zbl

[20] Guan P., Wang G., Geometric inequalities on locally conformally flat manifolds, Duke Math. J. 124 (2004) 177-212. | MR | Zbl

[21] Guan P., Wang G., A fully nonlinear conformal flow on locally conformally flat manifolds, math.DG/0112256, v1 of [19].

[22] Gursky M., Viaclovsky J., Volume comparison and the ${\sigma }_k$-Yamabe problem, Adv. in Math. 187 (2004) 447-487. | MR | Zbl

[23] Gursky M., Viaclovsky J., A fully nonlinear equation on 4-manifolds with positive scalar curvature, J. Differential Geom. 63 (2003) 131-154. | MR | Zbl

[24] Gursky M., Viaclovsky J., Prescribing symmetric functions of the eigenvalues of the Ricci tensor, Ann of Math., submitted for publication, math.DG/0409187.

[25] Habermann L., Riemannian Metrics of Constant Mass and Moduli Spaces of Conformal Structures, Lecture Notes in Mathematics, vol. 1743, Springer, Berlin, 2000. | MR | Zbl

[26] Hebey E., Nonlinear Analysis on Manifolds: Sobolev Spaces and Inequalities, Courant Lecture Notes in Math., vol. 5, Courant Inst. of Math. Sci./Amer. Math. Soc., New York/Providence, RI, 1999. | MR | Zbl

[27] Krylov N., Nonlinear Elliptic and Parabolic Equations of the Second Order, D. Reidel, Dordrecht, 1987. | MR | Zbl

[28] Lee J., Parker T., The Yamabe problem, Bull. Amer. Math. Soc. (N.S.) 17 (1987) 37-91. | MR | Zbl

[29] Li A., Li Y., On some conformally invariant fully nonlinear equations, Comm. Pure Appl. Math. 56 (2003) 1416-1464. | MR | Zbl

[30] Lions P.L., Two remarks on the Monge-Ampère, Ann. Mat. Pura Appl. 142 (1985) 263-275. | MR | Zbl

[31] Micallef M., Wang M., Metrics with nonnegative isotropic curvature, Duke Math. J. 72 (1993) 649-672. | MR | Zbl

[32] Schoen R., Conformal deformation of a Riemannian metric to constant curvature, J. Differential Geom. 20 (1984) 479-495. | MR | Zbl

[33] Sha J.-P., Yang D.G., Positive Ricci curvature on the connected sums of $S^n\times S^m$, J. Differential Geom. 33 (1991) 127-137. | MR | Zbl

[34] Sheng W., Trudinger N., Wang X., The Yamabe problem for higher order curvatures, math.DG/0505463.

[35] Simon L., Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems, Ann. of Math. 118 (1983) 525-571. | MR | Zbl

[36] Trudinger N., On imbeddings into Orlicz spaces and some applications, J. Math. Mech. 17 (1967) 473-483. | MR | Zbl

[37] Trudinger N., Wang X.-J., A Poincaré type inequality for Hessian integrals, Calc. Var. Partial Differential Equations 6 (4) (1998) 315-328. | MR | Zbl

[38] Viaclovsky J., Conformal geometry, contact geometry and the calculus of variations, Duke J. Math. 101 (2) (2000) 283-316. | MR | Zbl

[39] Viaclovsky J., Conformally invariant Monge-Ampère equations: Global solutions, Trans. Amer. Math. Soc. 352 (2000) 4371-4379. | MR | Zbl

[40] Viaclovsky J., Estimates and some existence results for some fully nonlinear elliptic equations on Riemannian manifolds, Comm. Anal. Geom. 10 (2002) 815-847. | MR | Zbl

[41] Wang X.J., A class of fully nonlinear elliptic equations and related functionals, Indiana Univ. Math. J. 43 (1994) 25-54. | MR | Zbl

[42] Yamabe H., On a deformation of Riemannian structures on compact manifolds, Osaka Math. J. 12 (1960) 21-37. | MR | Zbl