A priori estimates and existence for a class of fully nonlinear elliptic equations in conformal geometry

Chinese Annals of Mathematics. Series B

Volumn 27, Issue 2, 2006, Pages 169-178

  Wang, Xu Jia ^a  

^a AUSTRALIAN NATIONAL UNIVERSITY   (Australia)

Author keywords

Conformal geometry; Elliptic equation; Ricci curvature

Indexed keywords

EID: 33747885320     PISSN: 02529599     EISSN: 18606261     Source Type: Journal    
DOI: 10.1007/s11401-005-0529-6     Document Type: Article

References (23)

1. Aubin, T., Some Nonlinear Problems in Riemannian Geometry, Springer, 1998.
2. Caffarelli, L. A., Nirenberg, L. and Spruck, J., Dirichlet problem for nonlinear second order elliptic equations III. Functions of the eigenvalues of the Hessian, Acta Math., 155, 1985, 261-301.
3. Chang, A., Gursky, M. and Yang, P., An a priori estimate for a fully nonlinear equation on four-manifolds, J. Anal. Math., 87, 2002, 151-186.
4. Chen, S., Local estimates for some fully nonlinear elliptic equation, preprint, September, 2005.
5. Chou, K. S. and Wang, X.-J., A variational theory of the Hessian equation, Comm Purs Appl. Math., 54, 2001, 1029-1064.
6. Gerhardt, C., Closed Weingarten hypersurfaces in Riemannian manifolds, J. Diff. Geom., 43, 1996, 612-641.
7. Gilbarg, D. and Trudinger, N. S., Elliptic Partial Differential Equations of Second Order, Springer, 1983.
8. Guan, P. and Wang, G., Local estimates for a class of fully nonlinear equations arising from conformal geometry, Int. Math. Res. Not., 2003, 1413-1432.
9. Guan, P. and Wang, G., A fully nonlinear conformal flow on locally conformally flat manifolds, J. Reine Angew. Math., 557, 2003, 219-238.
10. Guan, P. and Wang, X.-J., On a Monge-Ampère equation arising in geometric optics, J. Diff. Geom., 48, 1998, 205-223.
11. Gursky, M. and Viaclovsky, J., Prescribing symmetric functions of the eigenvalues of the Ricci tensor, Ann. Math., to appear.
12. Gursky, M. and Viaclovsky, J., Convexity and singularities of curvature equations in conformal geometry, arXiv:math.DG/0504066.
13. Ivochkina, N., Solution of the Dirichlet problem for some equations of Monge-Ampère type, Mat. Sb., 128, 1985, 403-415.
14. Krylov, N. V., On the general notion of fully nonlinear second-order elliptic equations, Trans. Amer. Math. Soc., 347, 1995, 857-895.
15. Li, A. and Li, Y. Y., On some conformally invariant fully nonlinear equations, Comm. Pure Appl. Math., 56, 2003, 1416-1464.
16. Li, A. and Li, Y. Y., On some conformally invariant fully nonlinear equations II, Liouville, Harnack, and Yamabe, Acta Math., to appear.
17. Li, Y. Y., Degenerate conformal invariant fully nonlinear elliptic equations, arXiv:math.AP/0504598.
18. Sheng, W. M., Trudinger, N. S. and Wang, X.-J., The Yamabe problem for higher order curvatures, arXiv:math.DG/0505463.
19. Schoen, R. and Yau, S.-T., Lectures on Differential Geometry, International Press, 1994.
20. Trudinger, N. S. and Wang, X.-J., Hessian measures II, Ann. of Math., 150(2), 1999, 579-604.
21. Trudinger, N. S. and Wang, X.-J., On Harnack inequalities and singularities of admissible metrics in the Yamabe problem, arXiv:math.DG/ 0509341.
22. Viaclovsky, J., Estimates and existence results for some fully nonlinear elliptic equations on Riemannian manifolds, Comm. Anal. Geom., 10, 2002, 815-846.
23. Ye, R., Global existence and convergence of Yamabe flow, J. Diff. Geom., 39, 1994, 35-50.