On some conformally invariant fully nonlinear equations, II. Liouville, Harnack and Yamabe

Acta Mathematica

Volumn 195, Issue 1, 2005, Pages 117-154

  Li, Aobing ^a   Li, Yan Yan ^a  

^a RUTGERS UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 33644523132     PISSN: 00015962     EISSN: None     Source Type: Journal    
DOI: 10.1007/BF02588052     Document Type: Article

References (66)

1. AMBROSETTI, A., LI, Y. Y. & MALCHIODI, A., On the Yamabe problem and the scalar curvature problems under boundary conditions. Math. Ann., 322 (2002), 667-699.
2. AUBIN, T., Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire. J. Math. Pures Appl., 55 (1976), 269-296.
3. Some Nonlinear Problems in Riemannian Geometry. Springer, Berlin, 1998.
4. n/2. Calc. Var. Partial Differential Equations, 20 (2004), 399-402.
5. BRYANT, R., GRIFFITHS, P. & GROSSMAN, D., Exterior Differential Systems and Euler-Lagrange Partial Differential Equations. Univ. of Chicago Press, Chicago, IL, 2003.
6. CAFFARELLI, L. A., GIDAS, B. & SPRUCK, J., Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth. Comm. Pure Appl. Math., 42 (1989), 271-297.
7. 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.
8. CHANG, S.-Y. A., GURSKY, M. J. & YANG, P. C., An equation of Monge-Ampère type in conformal geometry, and four-manifolds of positive Ricci curvature. Ann. of Math., 155 (2002), 709-787.
9. An a priori estimate for a fully nonlinear equation on four-manifolds. J. Anal. Math., 87 (2002), 151-186.
10. A conformally invariant sphere theorem in four dimensions. Publ. Math. Inst. Hautes Études Sci., 98 (2003), 105-143.
11. Entire solutions of a fully nonlinear equation, in Lectures on Partial Differential Equations, pp. 43-60. New Stud. Adv. Math., 2. Int. Press, Sommerville, MA, 2003.
12. CHANG, S.-Y. A., HANG, F. & YANG, P. C., On a class of locally conformally flat manifolds. Int. Math. Res. Not., 2004:4 (2004), 185-209.
13. CHANG, S.-Y. A. & YANG, P. C., Non-linear partial differential equations in conformal geometry, in Proceedings of the International Congress of Mathematicians, Vol. I (Beijing, 2002), pp. 189-207. Higher Ed. Press, Beijing, 2002.
14. CHEN, W. X. & LI, C., Classification of solutions of some nonlinear elliptic equations. Duke Math. J., 63 (1991), 615-622.
15. DRUET, O., From one bubble to several bubbles: the low-dimensional case. J. Differential Geom., 63 (2003), 399-473.
16. EVANS, L. C., Classical solutions of fully nonlinear, convex, second-order elliptic equations. Comm. Pure Appl. Math., 35 (1982), 333-363.
17. FEFFERMAN, C. & GRAHAM, C.R., Conformai invariants, in The Mathematical Heritage of Élie Cartan (Lyon, 1984), pp. 95-116. Astérisque (numéro hors serie). Soc. Math. France, Paris, 1985.
18. GIDAS, B., NI, W. M. & NIRENBERG, L., Symmetry and related properties via the maximum principle. Comm. Math. Phys., 68 (1979), 209-243.
19. GONZALEZ, M., Ph.D. Thesis, Princeton University, 2004.
20. n for surfaces of constant Gauss curvature. Ann. of Math., 138 (1993), 601-624.
21. GUAN, P., LIN, C.-S. & WANG, G., Application of the method of moving planes to conformally invariant equations. Math. Z., 247 (2004), 1-19.
22. GUAN, P., VIACLOVSKY, J. & WANG, G., Some properties of the Scheuten tensor and applications to conformal geometry. Trans. Amer. Math. Soc., 355 (2003), 925-933.
23. GUAN, P. & WANG, G., Local estimates for a class of fully nonlinear equations arising from conformal geometry. Int. Math. Res. Not., 2003:26 (2003), 1413-1432.
24. A fully nonlinear conformal flow on locally conformally flat manifolds. J. Reine Angew. Math., 557 (2003), 219-238.
25. Geometric inequalities on locally conformally flat manifolds. Duke Math. J., 124 (2004), 177-212.
26. GURSKY, M. J., The principal eigenvalue of a conformally invariant differential operator, with an application to semilinear elliptic PDE. Comm. Math. Phys., 207 (1999), 131-143.
27. GURSKY, M. J. & VIACLOVSKY, J. A., A new variational characterization of three-dimensional space forms. Invent. Math., 145 (2001), 251-278.
28. Fully nonlinear equations on Riemannian manifolds with negative curvature. Indiana Univ. Math. J., 52 (2003), 399-419.
29. A fully nonlinear equation on four-manifolds with positive scalar curvature. J. Differential Geom., 63 (2003), 131-154.
30. κ-Yamabe problem. Adv. Math., 187 (2004), 447-487.
31. 2 curvature equation on 4-manifolds. Int. Math. Res. Not., 2004:79 (2004), 4269-4292.
32. KRYLOV, N. V., Boundedly inhomogeneous elliptic and parabolic equations in a domain. Izv. Akad. Nauk SSSR Ser. Mat., 47 (1983), 75-108 (Russian);
33. English translation in Math. USSR-Izv., 22 (1984), 67-98.
34. LI, A. & LI, Y. Y., On some conformally invariant fully nonlinear equations. C. R. Math. Acad. Sci. Paris, 334 (2002), 305-310.
35. A fully nonlinear version of the Yamabe problem and a Harnack type inequality. C. R. Math. Acad. Sci. Paris, 336 (2003), 319-324.
36. On some conformally invariant fully nonlinear equations. Comm. Pure Appl. Math., 56 (2003), 1416-1464.
37. A Liouville type theorem for some conformally invariant fully nonlinear equations, in Geometric Analysis of PDE and Several Complex Variables, pp. 321-328. Contemp. Math., 368. Amer. Math. Soc., Providence, RI, 2005.
38. A fully nonlinear version of the Yamabe problem and a Harnack type inequality. arXiv:math.AP/0212031.
39. A general Liouville type theorem for some conformally invariant fully nonlinear equations. arXiv:math.AP/0301239.
40. Further results on Liouville type theorems for some conformally invariant fully nonlinear equations. arXiv:math.AP/0301254.
41. On some conformally invariant fully nonlinear equations, II. Liouville, Harnack and Yamabe. arXiv:math.AP/0403442.
42. On some conformally invariant fully nonlinear equations, III. In preparation.
43. LI, Y. Y., Degree theory for second order nonlinear elliptic operators and its applications. Comm. Partial Differential Equations, 14 (1989), 1541-1578.
44. Some existence results of fully nonlinear elliptic equations of Monge-Ampère type. Comm. Pure Appl. Math., 43 (1990), 233-271.
45. On some conformally invariant fully nonlinear equations, in Proceedings of the International Congress of Mathematicians, Vol. Ill (Beijing, 2002), pp. 177-184. Higher Ed. Press, Beijing, 2002.
46. Liouville type theorems for some conformally invariant fully nonlinear equations. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat Appl., 14 (2003), 219-225.
47. LI, Y. Y. & ZHANG, L., Liouville-type theorems and Harnack-type inequalities for semilinear elliptic equations. J. Anal. Math., 90 (2003), 27-87.
48. Compactness of solutions to the Yamabe problem. C. R. Math. Acad. Sci. Paris, 338 (2004), 693-695.
49. A Harnack type inequality for the Yamabe equation in low dimensions. Calc. Var. Partial Differential Equations, 20 (2004), 133-151.
50. LI, Y. Y. & ZHU, M., Uniqueness theorems through the method of moving spheres. Duke Math. J., 80 (1995), 383-417.
51. Yamabe type equations on three-dimensional Riemannian manifolds. Commun. Contemp. Math., 1 (1999), 1-50.
52. OBATA, M., The conjectures on conformal transformations of Riemannian manifolds. J. Differential Geom., 6 (1971/72), 247-258.
53. ROCKAFELLAR, R. T., Convex Analysis. Reprint of the 1970 original. Princeton Univ. Press, Princeton, NJ, 1997.
54. SCHOEN, R., Conformal deformation of a Riemannian metric to constant scalar curvature. J. Differential Geom., 20 (1984), 479-495.
55. Courses at Stanford University, 1988, and at New York University, 1989.
56. On the number of constant scalar curvature metrics in a conformal class, in Differential Geometry, pp. 311-320. Pitman Monogr. Surveys Pure Appl. Math., 52. Longman Sci. Tech., Harlow, 1991.
57. SCHOEN, R. & YAU, S.-T., Conformally flat manifolds, Kleinian groups and scalar curvature. Invent. Math., 92 (1988), 47-71.
58. Lectures on Differential Geometry. International Press, Cambridge, MA, 1994.
59. TRUDINGER, N. S., Remarks concerning the conformal deformation of Riemannian structures on compact manifolds. Ann. Sc. Norm. Super. Pisa Cl. Sci., 22 (1968), 265-274.
60. On the Dirichlet problem for Hessian equations. Acta Math., 175 (1995), 151-164.
61. TRUDINGER, N.S. & WANG, X.-J., Hessian measures, II. Ann. of Math., 150 (1999), 579-604.
62. URBAS, J., Hessian equations on compact Riemannian manifolds, in Nonlinear Problems in Mathematical Physics and Related Topics, Vol. II, pp. 367-377. Int. Math. Ser. (N.Y.), 2. Kluwer/Plenum, New York, 2002.
63. VIACLOVSKY, J. A., Conformal geometry, contact geometry, and the calculus of variations. Duke Math. J., 101 (2000), 283-316.
64. Conformally invariant Monge-Ampère equations: global solutions. Trans. Amer. Math. Soc., 352 (2000), 4371-4379.
65. Estimates and existence results for some fully nonlinear elliptic equations on Riemannian manifolds. Comm. Anal. Geom., 10 (2002), 815-846.
66. YAMABE, H., On a deformation of Riemannian structures on compact manifolds. Osaka Math. J., 12 (1960), 21-37.