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Admissible metrics in the $ \sigma_{k}$-Yamabe equation

Author: Weimin Sheng
Journal: Proc. Amer. Math. Soc. 136 (2008), 1795-1802
MSC (2000): Primary 53C21; Secondary 53C20, 35J60
Published electronically: December 18, 2007
MathSciNet review: 2373610
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Abstract: In most previous works on the existence of solutions to the $ \sigma_{k} $-Yamabe problem, one assumes that the initial metric $ g_{0}$ is $ k$-admissible. This is a pointwise condition. In this paper we prove that this condition can be replaced by a weaker integral condition.

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  • 1. T. Aubin, Equations differentielles non lineaires et problème de Yamabe concernant la courbure scalaire, J. Math. Pures Appl., 55 (1976), 269-296. MR 0431287 (55:4288)
  • 2. Arthur L. Besse, Einstein manifolds, Springer-Verlag, Berlin, 1987. MR 867684 (88f:53087)
  • 3. S. Brendle and J. Viaclovsky, A variational characterization for $ \sigma_{n/2}$, Calc. Var. PDE., 20 (2004), 399-402. MR 2071927 (2005c:53081)
  • 4. L. Caffarelli, L. Nirenberg, and J. Spruck, The Dirichlet problem for nonlinear second order elliptic equations, III: Functions of the eigenvalues of the Hessian, Acta Math., 155 (1985), 261-301. MR 806416 (87f:35098)
  • 5. A. Chang, M. Gursky, and P. Yang, An equation of Monge-Ampère type in conformal geometry, and four-manifolds of positive Ricci curvature, Ann. of Math. (2), 155 (2002), 709-787. MR 1923964 (2003j:53048)
  • 6. A. Chang, M. Gursky, and P. Yang, An a priori estimate for a fully nonlinear equation on four-manifolds, J. Anal. Math., 87 (2002), 151-186. MR 1945280 (2003k:53036)
  • 7. S. Chen, Local estimates for some fully nonlinear elliptic equations, arXiv: math.AP/0510652.
  • 8. Y. Ge and G. Wang, On a fully nonlinear Yamabe problem, preprint. MR 2290138
  • 9. D. Gilbarg and N.S. Trudinger, Elliptic partial differential equations of second order, Springer, Berlin, 1983. MR 737190 (86c:35035)
  • 10. P. Guan, C.-S. Lin, and G. Wang, Application of the method of moving planes to conformally invariant equations, Math. Z., 247 (2004), 1-19. MR 2054518 (2005a:53059)
  • 11. P. Guan and G. Wang, Local estimates for a class of conformal equation arising from conformal geometry, Int. Math. Res. Not., 2003 (2003), 1413-1432. MR 1976045 (2003m:53055)
  • 12. P. Guan and G. Wang, A fully nonlinear conformal flow on locally conformally flat manifolds, J. Reine Angew. Math., 557 (2003), 219-238. MR 1978409 (2004e:53101)
  • 13. M. Gursky and J. Viaclovsky, Fully nonlinear equations on Riemannian manifolds with negative curvature, Indiana Univ. Math. J., 52 (2003), 399-419. MR 1976082 (2004a:53039)
  • 14. M. Gursky and J. Viaclovsky, A fully nonlinear equation on four-manifolds with positive scalar curvature, J. Differential Geom., 63 (2003), 131-154. MR 2015262 (2004h:53052)
  • 15. M. Gursky and J. Viaclovsky, Prescribing symmetric functions of the eigenvalues of the Ricci tensor, Ann. of Math., to appear.
  • 16. N.V. Krylov, Nonlinear elliptic and parabolic equations of second order, Reidel, 1987.
  • 17. N.V. Krylov, On the general notion of fully nonlinear second-order elliptic equations, Trans. Amer. Math. Soc., 347 (1995), 857-895. MR 1284912 (95f:35075)
  • 18. A. Li and Y.Y. Li, On some conformally invariant fully nonlinear equations, Comm. Pure Appl. Math., 56 (2003), 1416-1464. MR 1988895 (2004e:35072)
  • 19. J. Li and W.M. Sheng, Deforming metrics with negative curvature by a fully nonlinear flow, Calc. Var. PDE., 23 (2005), 33-50. MR 2133660 (2005m:53121)
  • 20. R. Schoen, Conformal deformation of a Riemannian metric to constant scalar curvature, J. Differential Geom., 20 (1984), 479-495. MR 788292 (86i:58137)
  • 21. W.M. Sheng, N. Trudinger and X.-J. Wang, The Yamabe problem for higher order curvatures, J. Differential Geom. 77 (2007), 515-553.
  • 22. W.M. Sheng, J. Urbas, and X.-J. Wang, Interior curvature bounds for a class of curvature equations, Duke Math. J., 123 (2004), 235-264. MR 2066938 (2005d:35087)
  • 23. W.M. Sheng and Y. Zhang, A class of fully nonlinear equations arising conformal geometry, Math. Z., 255 (2007), 17-34. MR 2262720
  • 24. N. Trudinger, On Harnack type inequalities and their application to quasilinear elliptic equations, Comm. Pure Appl. Math., 20 (1967), 721-747. MR 0226198 (37:1788)
  • 25. N.S. Trudinger, The Dirichlet problem for the prescribed curvature equations, Arch. Rational Mech. Anal., 111 (1990), 153-179. MR 1057653 (91g:35118)
  • 26. N. Trudinger and X.-J. Wang, On Harnack inequalities and singularities of admissible metrics in the Yamabe problem, arXiv:math.DG/0509341.
  • 27. J. Viaclovsky, Conformal geometry, contact geometry, and the calculus of variations, Duke Math. J., 101 (2000), 283-316. MR 1738176 (2001b:53038)
  • 28. X.-J. Wang, A priori estimates and existence for a class of fully nonlinear elliptic equations in conformal geometry, Chin. Ann. of Math. (B), 27 (2006), 169-178. MR 2243678 (2007e:53036)

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Additional Information

Weimin Sheng
Affiliation: Department of Mathematics, Zhejiang University, Hangzhou 310027, People’s Republic of China

Keywords: $\sigma_{k}$-curvature, admissible metrics, $k$-Yamabe constant
Received by editor(s): January 30, 2007
Received by editor(s) in revised form: February 21, 2007
Published electronically: December 18, 2007
Additional Notes: The author was supported in part by NSFC Grant #10471122.
Communicated by: Richard A. Wentworth
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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