On bifurcation of solutions of the Yamabe problem in product manifolds
Annales de l'I.H.P. Analyse non linéaire, Tome 29 (2012) no. 2, pp. 261-277.

We study local rigidity and multiplicity of constant scalar curvature metrics in arbitrary products of compact manifolds. Using (equivariant) bifurcation theory we determine the existence of infinitely many metrics that are accumulation points of pairwise non-homothetic solutions of the Yamabe problem. Using local rigidity and some compactness results for solutions of the Yamabe problem, we also exhibit new examples of conformal classes (with positive Yamabe constant) for which uniqueness holds.

DOI : 10.1016/j.anihpc.2011.10.005
Classification : 58E11, 58J55, 58E09
@article{AIHPC_2012__29_2_261_0,
     author = {de Lima, L.L. and Piccione, P. and Zedda, M.},
     title = {On bifurcation of solutions of the {Yamabe} problem in product manifolds},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {261--277},
     publisher = {Elsevier},
     volume = {29},
     number = {2},
     year = {2012},
     doi = {10.1016/j.anihpc.2011.10.005},
     mrnumber = {2901197},
     zbl = {1239.58005},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2011.10.005/}
}
TY  - JOUR
AU  - de Lima, L.L.
AU  - Piccione, P.
AU  - Zedda, M.
TI  - On bifurcation of solutions of the Yamabe problem in product manifolds
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2012
SP  - 261
EP  - 277
VL  - 29
IS  - 2
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.anihpc.2011.10.005/
DO  - 10.1016/j.anihpc.2011.10.005
LA  - en
ID  - AIHPC_2012__29_2_261_0
ER  - 
%0 Journal Article
%A de Lima, L.L.
%A Piccione, P.
%A Zedda, M.
%T On bifurcation of solutions of the Yamabe problem in product manifolds
%J Annales de l'I.H.P. Analyse non linéaire
%D 2012
%P 261-277
%V 29
%N 2
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.anihpc.2011.10.005/
%R 10.1016/j.anihpc.2011.10.005
%G en
%F AIHPC_2012__29_2_261_0
de Lima, L.L.; Piccione, P.; Zedda, M. On bifurcation of solutions of the Yamabe problem in product manifolds. Annales de l'I.H.P. Analyse non linéaire, Tome 29 (2012) no. 2, pp. 261-277. doi : 10.1016/j.anihpc.2011.10.005. http://www.numdam.org/articles/10.1016/j.anihpc.2011.10.005/

[1] M.T. Anderson, On uniqueness and differentiability in the space of Yamabe metrics, Commun. Contemp. Math. 7 no. 3 (2005), 299-310 | MR | Zbl

[2] T. Aubin, Équations différentielles non-linéaires et problème de Yamabe concernant la courbure scalaire, J. Math. Pures Appl. 55 (1976), 269-296 | MR | Zbl

[3] M. Berger, P. Gauduchon, E. Mazet, Le Spectre dʼune Variété Riemannienne, Lecture Notes in Math. vol. 194, Springer-Verlag, Berlin, Heidelberg, New York (1971) | MR

[4] A.L. Besse, Einstein Manifolds, Classics Math., Springer-Verlag, Berlin (2008) | MR | Zbl

[5] C. Böhm, M. Wang, W. Ziller, A variational approach for compact homogeneous Einstein manifolds, Geom. Funct. Anal. 14 no. 4 (2004), 681-733 | MR | Zbl

[6] E. Hebey, M. Vaugon, Meilleures constantes dans le théorème dʼinclusion de Sobolev et multiplicité pour les problèmes de Nirenberg et Yamabe, Indiana Univ. Math. J. 41 no. 2 (1992), 377-407 | MR

[7] Q. Jin, Y. Li, H. Xu, Symmetry and asymmetry: the method of moving spheres, Adv. Differential Equations 13 no. 7–8 (2008), 601-640 | MR | Zbl

[8] M.A. Khuri, F.C. Marques, R.M. Schoen, A compactness theorem for the Yamabe problem, J. Differential Geom. 81 no. 1 (2009), 143-196 | MR | Zbl

[9] O. Kobayashi, Scalar curvature of a metric with unit volume, Math. Ann. 279 no. 2 (1987), 253-265 | EuDML | MR | Zbl

[10] N. Koiso, On the second derivative of the total scalar curvature, Osaka J. Math. 16 (1979), 413-421 | MR | Zbl

[11] C. Lebrun, Einstein metrics and the Yamabe problem, Trends in Mathematical Physics, Knoxville, TN, 1998, AMS/IP Stud. Adv. Math. vol. 13, Amer. Math. Soc., Providence, RI (1999), 353-376 | MR | Zbl

[12] Y.Y. Li, L. Zhang, Compactness of solutions to the Yamabe problem. II, Calc. Var. Partial Differential Equations 24 no. 2 (2005), 185-237 | MR | Zbl

[13] F.C. Marques, A priori estimates for the Yamabe problem in the non-locally conformally flat case, J. Differential Geom. 71 no. 2 (2005), 315-346 | MR | Zbl

[14] M. Obata, Certain conditions for a Riemannian manifold to be isometric with a sphere, J. Math. Soc. Japan 14 no. 3 (1962), 333-340 | MR | Zbl

[15] M. Obata, The conjectures on conformal transformations of Riemannian manifolds, J. Differential Geom. (1971), 247-258 | MR | Zbl

[16] J. Petean, Metrics of constant scalar curvature conformal to Riemannian products, Proc. Amer. Math. Soc. 138 no. 8 (2010), 2897-2905 | MR | Zbl

[17] D. Pollack, Nonuniqueness and high energy solutions for a conformally invariant scalar equation, Comm. Anal. Geom. 1 no. 3–4 (1993), 347-414 | MR | Zbl

[18] R. Schoen, Conformal deformation of a Riemannian metric to constant scalar curvature, J. Differential Geom. 20 (1984), 479-495 | MR | Zbl

[19] R. Schoen, Variational theory for the total scalar curvature functional for Riemannian metrics and related topics, Topics in Calculus of Variations, Lecture Notes in Math. vol. 1365 (1989), 120-154 | MR

[20] J. Smoller, A.G. Wasserman, Bifurcation and symmetry-breaking, Invent. Math. 100 (1990), 63-95 | EuDML | MR | Zbl

[21] N. Trudinger, Remarks concerning the conformal deformation of Riemannian structures on compact manifolds, Ann. Sc. Norm. Super. Pisa 22 (1968), 265-274 | EuDML | Numdam | MR | Zbl

[22] H. Yamabe, On a deformation of Riemannian structures on compact manifolds, Osaka J. Math. 12 (1960), 21-37 | MR | Zbl

[23] S.T. Yau, Remarks on conformal transformations, J. Differential Geom. 8 (1973), 369-381 | MR | Zbl

[24] B. White, The space of m-dimensional surfaces that are stationary for a parametric elliptic functional, Indiana Univ. Math. J. 36 (1987), 567-602 | MR | Zbl

[25] B. White, The space of minimal submanifolds for varying Riemannian metrics, Indiana Univ. Math. J. 40 (1991), 161-200 | MR | Zbl

Cité par Sources :