Remote Access Journal of the American Mathematical Society
Green Open Access

Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)

 
 

 

Blow-up phenomena for the Yamabe equation


Author: Simon Brendle
Journal: J. Amer. Math. Soc. 21 (2008), 951-979
MSC (2000): Primary 53C21; Secondary 53C44
DOI: https://doi.org/10.1090/S0894-0347-07-00575-9
Published electronically: June 14, 2007
MathSciNet review: 2425176
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ (M,g)$ be a compact Riemannian manifold of dimension $ n \geq 3$. A well-known conjecture states that the set of constant scalar curvature metrics in the conformal class of $ g$ is compact unless $ (M,g)$ is conformally equivalent to the round sphere. In this paper, we construct counterexamples to this conjecture in dimensions $ n \geq 52$.


References [Enhancements On Off] (What's this?)

  • 1. A. Ambrosetti, Multiplicity results for the Yamabe problem on $ S^n$, Proc. Natl. Acad. Sci. USA 99 (2002), 15252-15256. MR 1946759 (2003j:53047)
  • 2. A. Ambrosetti and A. Malchiodi, A multiplicity result for the Yamabe problem on $ S^n$, J. Funct. Anal. 168, 529-561 (1999). MR 1719213 (2000k:53032)
  • 3. T. Aubin, Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire, J. Math. Pures Appl. 55, 269-296 (1976). MR 0431287 (55:4288)
  • 4. T. Aubin, Sur quelques problèmes de courbure scalaire, J. Funct. Anal. 240, 269-289 (2006). MR 2259897
  • 5. T. Aubin, Solution complète de la $ C^0$ compacité de l'ensemble des solutions de l'équation de Yamabe, J. Funct. Anal. 244, 579-589 (2007). MR 2297036
  • 6. M. Berti and A. Malchiodi, Non-compactness and multiplicity results for the Yamabe problem on $ S^n$, J. Funct. Anal. 180, 210-241 (2001). MR 1814428 (2002b:53049)
  • 7. O. Druet, Compactness for Yamabe metrics in low dimensions, Internat. Math. Res. Notices 23, 1143-1191 (2004). MR 2041549 (2005b:53056)
  • 8. O. Druet and E. Hebey, Blow-up examples for second order elliptic PDEs of critical Sobolev growth, Trans. Amer. Math. Soc. 357, 1915-1929 (2004). MR 2115082 (2005i:58023)
  • 9. O. Druet and E. Hebey, Elliptic equations of Yamabe type, International Mathematics Research Surveys 1, 1-113 (2005). MR 2148873 (2006b:53046)
  • 10. M. Khuri, F. Marques, and R. Schoen, A compactness theorem for the Yamabe problem, preprint (2007).
  • 11. Y.Y. Li and L. Zhang, Compactness of solutions to the Yamabe problem II, Calc. Var. PDE 24, 185-237 (2005). MR 2164927 (2006f:53049)
  • 12. Y.Y. Li and M. Zhu, Yamabe type equations on three-dimensional Riemannian manifolds, Commun. Contemp. Math. 1, 1-50 (1999). MR 1681811 (2000m:53051)
  • 13. F.C. Marques, A-priori estimates for the Yamabe problem in the non-locally conformally flat case, J. Diff. Geom. 71, 315-346 (2005). MR 2197144 (2006i:53046)
  • 14. D. Pollack, Nonuniqueness and high energy solutions for a conformally invariant scalar equation, Comm. Anal. Geom. 1, 347-414 (1993). MR 1266473 (94m:58051)
  • 15. O. Rey, The role of the Green's function in a non-linear elliptic equation involving the critical Sobolev exponent, J. Funct. Anal. 89, 1-52 (1990). MR 1040954 (91b:35012)
  • 16. R.M. Schoen, Conformal deformation of a Riemannian metric to constant scalar curvature, J. Diff. Geom. 20, 479-495 (1984) MR 788292 (86i:58137)
  • 17. R.M. Schoen, Variational theory for the total scalar curvature functional for Riemannian metrics and related topics, Topics in the calculus of variations (ed. by Mariano Giaquinta), Lecture Notes in Mathematics, vol. 1365, Springer Verlag, 1989, 120-154. MR 994021 (90g:58023)
  • 18. R.M. Schoen, On the number of constant scalar curvature metrics in a conformal class, Differential geometry (ed. by H. Blaine Lawson, Jr., and Keti Tenenblat), Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 52, Longman Scientific & Technical, 1991, 311-320. MR 1173050 (94e:53035)
  • 19. R.M. Schoen, A report on some recent progress on nonlinear problems in geometry, In: Surveys in differential geometry, Lehigh University, Bethlehem, PA, 1991, 201-241. MR 1144528 (92m:53069)
  • 20. N. Trudinger, Remarks concerning the conformal deformation of Riemannian structures on compact manifolds, Annali Scuola Norm. Sup. Pisa 22, 265-274 (1968). MR 0240748 (39:2093)

Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (2000): 53C21, 53C44

Retrieve articles in all journals with MSC (2000): 53C21, 53C44


Additional Information

Simon Brendle
Affiliation: Department of Mathematics, Stanford University, Stanford, California 94305

DOI: https://doi.org/10.1090/S0894-0347-07-00575-9
Keywords: Scalar curvature, conformal deformation of Riemannian metrics, blow-up analysis
Received by editor(s): October 23, 2006
Published electronically: June 14, 2007
Additional Notes: This project was supported by the Alfred P. Sloan Foundation and by the National Science Foundation under grant DMS-0605223.
Article copyright: © Copyright 2007 American Mathematical Society

American Mathematical Society