Admissible metrics in the 𝜎_{𝑘}-Yamabe equation

Sheng, Weimin

doi:10.1090/S0002-9939-07-09167-8

Admissible metrics in the $\sigma _{k}$-Yamabe equation
HTML articles powered by AMS MathViewer

by Weimin Sheng PDF

Proc. Amer. Math. Soc. 136 (2008), 1795-1802 Request permission

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.

References

Thierry Aubin, Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire, J. Math. Pures Appl. (9) 55 (1976), no. 3, 269–296. MR 431287
Arthur L. Besse, Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 10, Springer-Verlag, Berlin, 1987. MR 867684, DOI 10.1007/978-3-540-74311-8
Simon Brendle and Jeff A. Viaclovsky, A variational characterization for $\sigma _{n/2}$, Calc. Var. Partial Differential Equations 20 (2004), no. 4, 399–402. MR 2071927, DOI 10.1007/s00526-003-0234-9
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), no. 3-4, 261–301. MR 806416, DOI 10.1007/BF02392544
Sun-Yung A. Chang, Matthew J. Gursky, and Paul C. Yang, An equation of Monge-Ampère type in conformal geometry, and four-manifolds of positive Ricci curvature, Ann. of Math. (2) 155 (2002), no. 3, 709–787. MR 1923964, DOI 10.2307/3062131
Sun-Yung A. Chang, Matthew J. Gursky, and Paul Yang, An a priori estimate for a fully nonlinear equation on four-manifolds, J. Anal. Math. 87 (2002), 151–186. Dedicated to the memory of Thomas H. Wolff. MR 1945280, DOI 10.1007/BF02868472
S. Chen, Local estimates for some fully nonlinear elliptic equations, arXiv: math.AP/0510652.
Yuxin Ge and Guofang Wang, On a fully nonlinear Yamabe problem, Ann. Sci. École Norm. Sup. (4) 39 (2006), no. 4, 569–598 (English, with English and French summaries). MR 2290138, DOI 10.1016/j.ansens.2005.12.007
David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR 737190, DOI 10.1007/978-3-642-61798-0
Pengfei Guan, Chang-Shou Lin, and Guofang Wang, Application of the method of moving planes to conformally invariant equations, Math. Z. 247 (2004), no. 1, 1–19. MR 2054518, DOI 10.1007/s00209-003-0608-x
Pengfei Guan and Guofang Wang, Local estimates for a class of fully nonlinear equations arising from conformal geometry, Int. Math. Res. Not. 26 (2003), 1413–1432. MR 1976045, DOI 10.1155/S1073792803212034
Pengfei Guan and Guofang Wang, A fully nonlinear conformal flow on locally conformally flat manifolds, J. Reine Angew. Math. 557 (2003), 219–238. MR 1978409, DOI 10.1515/crll.2003.033
Matthew J. Gursky and Jeff A. Viaclovsky, Fully nonlinear equations on Riemannian manifolds with negative curvature, Indiana Univ. Math. J. 52 (2003), no. 2, 399–419. MR 1976082, DOI 10.1512/iumj.2003.52.2313
Matthew J. Gursky and Jeff A. Viaclovsky, A fully nonlinear equation on four-manifolds with positive scalar curvature, J. Differential Geom. 63 (2003), no. 1, 131–154. MR 2015262
M. Gursky and J. Viaclovsky, Prescribing symmetric functions of the eigenvalues of the Ricci tensor, Ann. of Math., to appear.
N.V. Krylov, Nonlinear elliptic and parabolic equations of second order, Reidel, 1987.
N. V. Krylov, On the general notion of fully nonlinear second-order elliptic equations, Trans. Amer. Math. Soc. 347 (1995), no. 3, 857–895. MR 1284912, DOI 10.1090/S0002-9947-1995-1284912-8
Aobing Li and Yanyan Li, On some conformally invariant fully nonlinear equations, Comm. Pure Appl. Math. 56 (2003), no. 10, 1416–1464. MR 1988895, DOI 10.1002/cpa.10099
Jiayu Li and Weimin Sheng, Deforming metrics with negative curvature by a fully nonlinear flow, Calc. Var. Partial Differential Equations 23 (2005), no. 1, 33–50. MR 2133660, DOI 10.1007/s00526-004-0287-4
Richard Schoen, Conformal deformation of a Riemannian metric to constant scalar curvature, J. Differential Geom. 20 (1984), no. 2, 479–495. MR 788292
W.M. Sheng, N. Trudinger and X.-J. Wang, The Yamabe problem for higher order curvatures, J. Differential Geom. 77 (2007), 515-553.
Weimin Sheng, John Urbas, and Xu-Jia Wang, Interior curvature bounds for a class of curvature equations, Duke Math. J. 123 (2004), no. 2, 235–264. MR 2066938, DOI 10.1215/S0012-7094-04-12321-8
Weimin Sheng and Yan Zhang, A class of fully nonlinear equations arising from conformal geometry, Math. Z. 255 (2007), no. 1, 17–34. MR 2262720, DOI 10.1007/s00209-006-0011-5
Neil S. Trudinger, On Harnack type inequalities and their application to quasilinear elliptic equations, Comm. Pure Appl. Math. 20 (1967), 721–747. MR 226198, DOI 10.1002/cpa.3160200406
Neil S. Trudinger, The Dirichlet problem for the prescribed curvature equations, Arch. Rational Mech. Anal. 111 (1990), no. 2, 153–179. MR 1057653, DOI 10.1007/BF00375406
N. Trudinger and X.-J. Wang, On Harnack inequalities and singularities of admissible metrics in the Yamabe problem, arXiv:math.DG/0509341.
Jeff A. Viaclovsky, Conformal geometry, contact geometry, and the calculus of variations, Duke Math. J. 101 (2000), no. 2, 283–316. MR 1738176, DOI 10.1215/S0012-7094-00-10127-5
Xu-Jia Wang, A priori estimates and existence for a class of fully nonlinear elliptic equations in conformal geometry, Chinese Ann. Math. Ser. B 27 (2006), no. 2, 169–178. MR 2243678, DOI 10.1007/s11401-005-0529-6