The yamabe problem for higher order curvatures

Journal of Differential Geometry

Volumn 77, Issue 3, 2007, Pages 515-553

  Sheng, Wei Min ^a   Trudinger, Neil S ^b   Wang, Xu Jia ^b,c  

^a ZHEJIANG UNIVERSITY   (China)

^b AUSTRALIAN NATIONAL UNIVERSITY   (Australia)

^c NANKAI UNIVERSITY   (China)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 36849036198     PISSN: 0022040X     EISSN: 1945743X     Source Type: Journal    
DOI: 10.4310/jdg/1193074903     Document Type: Article

References (54)

1. T. Aubin, Equations différentielles non linéaires et problme de Yamabe concernant la courbure scalaire, J. Math. Pures Appl. (9) 55 (1976) 269-296.
2. T. Branson & A.R. Gover, Variational status of a class of fully nonlinear curvature prescription problems, preprint.
3. S. Brendle, Blow-up phenomena for the Yamabe PDE in high dimensions, preprint.
4. S. Brendle & F.C. Marques, Blow-up phenomena for the Yamabe problem II, preprint.
5. 2, Calc. Var. PDE. 20(2004), 399-402.
6. L.A. Caffarelli, L. Nirenberg, & J. Spruck, Dirichlet problem for nonlinear second order elliptic equations III, Functions of the eigenvalues of the Hessian, Acta Math. 1985(l55) 261-301.
7. A. Chang & P. Yang, The inequality of Moser and Trudinger and applications to conformal geometry, Comm. Pure Appl. Math. 56 (2003) 1135-1150.
8. A. Chang, M. Gursky, & P. Yang, An equation of Monge-Amère type in conformal geometry, and four-manifolds of positive Ricci curvature, Ann. of Math. (2) 155 (2002) 709-787.
9. A. Chang, M. Gursky, & P. Yang, An a priori estimate for a fully nonlinear equation on four-manifolds, J. Anal. Math. 87 (2002), 151-186.
10. A. Chang, M. Gursky, & P. Yang, Entire solutions of a fully nonlinear equation, Lectures on partial differential equations, 43-60, Inter. Press, 2003.
11. S. Chen, Local estimates for some fully nonlinear elliptic equations, Int. Math. Res. Not. 55 (2005) 3403-3425.
12. K.S. Chou (K. Tso), On a real Monge-Ampere functional, Invent. Math. 101 (1990) 425-448.
13. K.S. Chou & X-J. Wang, Variational solutions to Hessian equations, preprint, 1996, available at wwwmaths.anu.edu.au/research.reports.
14. K.S. Chou & X-J. Wang, A variational theory of the Hessian equation, Comm. Pure Appl. Math. 54 (2001) 1029-1064.
15. O. Druet, Compactness for Yamabe metrics in low dimensions, Int. Math. Res. Notes 23 (2004) 1143-1191.
16. L.C. Evans, A convergence theorem for solutions of nonlinear second-order elliptic equations, Indiana Univ. Math. J. 27 (1978) 875-887.
17. Y. Ge & G. Wang, On a fully nonlinear Yamabe problem, Ann. Sci. Ecole Norm. Sup. (4) 39 (2006) 569-598.
18. D. Gilbarg & N.S. Trudinger, Elliptic partial differential equations of second order, Springer, 1983.
19. P. Guan & G. Wang, Local estimates for a class of fully nonlinear equations arising from conformal geometry, Int. Math. Res. Not., 2003, 1413-1432.
20. P. Guan & G. Wang, A fully nonlinear conformal flow on locally conformally flat manifolds, J. Reine Angew. Math. 557 (2003) 219-238.
21. P. Guan & G. Wang, Geometric inequalities on locally conformally flat manifolds, Duke Math. J. 124 (2004) 177-212.
22. P. Guan & G. Wang, On a Monge-Ampere equation arising in geometric optics, J. Differential Geom. 48 (1998) 205-223.
23. M. Gursky & J. Viaclovsky, A fully nonlinear equation on four-manifolds with positive scalar curvature, J. Differential Geom. 63 (2003) 131-154.
24. M. Gursky & J. Viaclovsky, Prescribing symmetric functions of the eigenvalues of the Ricci tensor, arXiv:math.DG/0409187, Ann. Math., to appear.
25. M.A. Khuri, F.C. Marques, & R.M. Schoen, A compactness theorem for the Yamabe problem, preprint.
26. N. Ivochkina, Solution of the Dirichlet problem for certain equations of Monge-Ampere type (Russian), Mat. Sb. (N.S.) 128(170) (1985) 403-415.
27. M.-L. Labbi, Variational properties of the Gauss-Bonnet curvatures, preprint.
28. A. Li & Y.Y. Li, On some conformally invariant fully nonlinear equations, Comm. Pure Appl. Math. 56 (2003) 1416-1464.
29. A. Li & Y.Y. Li, On some conformally invariant fully nonlinear equations II, Liou-ville, Harnack, and Yamabe, Acta Math. 195 (2005) 117-154.
30. Y.Y. Li, Local gradient estimates of solutions to some conformally invariant fully nonlinear equations, C.R. Math. Acad. Sci. Paris 343 (2006) 249-252.
31. Y.Y. Li & L. Zhang, Compactness of solutions to the Yamabe problem II, Calc. Var. PDE 24 (2005) 185-237.
32. Y.Y. Li & L. Zhang, Compactness of solutions to the Yamabe problem III, J. Funct. Anal. 245 (2007) 438-474.
33. M. Lin & N.S. Trudinger, On some inequalities for elementary symmetric functions, Bull. Aust. Math. Soc. 50 (1994) 317-326.
34. X.N. Ma, N.S. Trudinger, & X-J. Wang, Regularity of potential functions to the optimal transportation problem, Arch. Ration. Mech. Anal. 177 (2005) 151-183.
35. F.C. Marques, A priori estimates for the Yamabe problem in non-locally conformal flat cases, arXiv:math.DG/0408063.
36. P.J. Olver, Applications of Lie groups to differential equations, Springer, 1993.
37. k Yamabe equation, Proc. Amer. Math. Soc., to appear.
38. R. Schoen, Conformal deformation of a Riemannian metric to constant scalar curvature, J. Differential Geom. 20 (1984) 479-495.
39. R. Schoen, Variational theory for the total scalar curvature functional for Riemannian metrics and related topics, in ‘Topics in Calculus of Variations’, Lectures Notes in Math., 1365, Springer, 1989, 120-154.
40. R. Schoen & S.-T. Yau, Conformally flat manifolds, Kleinian groups and scalar curvature, Invent. Math. 92 (1988) 47-71.
41. R. Schoen & S.-T. Yau, Lectures on Differential geometry, International Press, 1994.
42. F. Schulz, Regularity theory for quasi-linear elliptic systems and MongeAmpere equations in two dimensions, Lecture Notes in Math., 1445, 1990.
43. N.S. Trudinger, Remarks concerning the conformal deformation of Riemann-ian structures on compact manifolds, Ann. Scuola Norm. Sup. Pisa (3) 22 (1968) 265-274.
44. N.S. Trudinger and X-J. Wang, Hessian measures I, Ann. of Math. (2) 150 (1999) 579-604.
45. N.S. Trudinger and X-J. Wang, A Poincare type inequality for Hessian integrals, Calc. Var. PDE 6 (1998) 315-328.
46. N.S. Trudinger and X-J. Wang, On Harnack inequalities and singularities of admissible metrics in the Yamabe problem, preprint.
47. J. Viaclovsky, Conformal geometry, contact geometry, and the calculus of variations, Duke Math. J. 101 (2000) 283-316.
48. J. Viaclovsky, Estimates and existence results for some fully nonlinear elliptic equations on Riemannian manifolds, Comm. Anal. Geom. 10 (2002) 815-846.
49. X.-J. Wang, A class of fully nonlinear elliptic equations and related functionals, Indiana Univ. Math. J. 43 (1994) 25-54.
50. X.-J. Wang, A note on the Yamabe problem, J. Partial Diff. Eqns 18 (2005) 322-326.
51. X.-J. Wang, A priori estimates and existence for a class of fully nonlinear elliptic equations in conformal geometry, Chinese Ann. Math. Ser. B 27 (2006) 169-178.
52. X.-J. Wang, The k-Hessian equation, Lect. Notes Math., Springer, to appear.
53. H. Yamabe, On a deformation of Riemannian structures on compact manifolds, Osaka Math. J. 12 (1960) 21-37.
54. R. Ye, Global existence and convergence of Yamabe flow, J. Differential Geom. 39 (1994) 35-50.