On bifurcation of solutions of the Yamabe problem in product manifolds
Annales de l'I.H.P. Analyse non linéaire, Volume 29 (2012) no. 2, p. 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 : https://doi.org/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},
     publisher = {Elsevier},
     volume = {29},
     number = {2},
     year = {2012},
     pages = {261-277},
     doi = {10.1016/j.anihpc.2011.10.005},
     zbl = {1239.58005},
     mrnumber = {2901197},
     language = {en},
     url = {http://www.numdam.org/item/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, Volume 29 (2012) no. 2, pp. 261-277. doi : 10.1016/j.anihpc.2011.10.005. http://www.numdam.org/item/AIHPC_2012__29_2_261_0/

[1] M.T. Anderson, On uniqueness and differentiability in the space of Yamabe metrics, Commun. Contemp. Math. 7 no. 3 (2005), 299-310 | MR 2151861 | Zbl 1082.58013

[2] T. Aubin, Équations différentielles non-linéaires et problème de Yamabe concernant la courbure scalaire, J. Math. Pures Appl. 55 (1976), 269-296 | MR 431287 | Zbl 0336.53033

[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 282313

[4] A.L. Besse, Einstein Manifolds, Classics Math., Springer-Verlag, Berlin (2008) | MR 2371700 | Zbl 1147.53001

[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 2084976 | Zbl 1068.53029

[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 1183349

[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 2479025 | Zbl 1201.35099

[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 2477893 | Zbl 1162.53029

[9] O. Kobayashi, Scalar curvature of a metric with unit volume, Math. Ann. 279 no. 2 (1987), 253-265 | MR 919505 | Zbl 0611.53037

[10] N. Koiso, On the second derivative of the total scalar curvature, Osaka J. Math. 16 (1979), 413-421 | MR 539596 | Zbl 0415.53033

[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 1708770 | Zbl 1050.53032

[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 2164927 | Zbl 1229.35071

[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 2197144 | Zbl 1101.53019

[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 142086 | Zbl 0115.39302

[15] M. Obata, The conjectures on conformal transformations of Riemannian manifolds, J. Differential Geom. (1971), 247-258 | MR 303464 | Zbl 0236.53042

[16] J. Petean, Metrics of constant scalar curvature conformal to Riemannian products, Proc. Amer. Math. Soc. 138 no. 8 (2010), 2897-2905 | MR 2644902 | Zbl 1206.53042

[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 1266473 | Zbl 0848.58011

[18] R. Schoen, Conformal deformation of a Riemannian metric to constant scalar curvature, J. Differential Geom. 20 (1984), 479-495 | MR 788292 | Zbl 0576.53028

[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 994021

[20] J. Smoller, A.G. Wasserman, Bifurcation and symmetry-breaking, Invent. Math. 100 (1990), 63-95 | MR 1037143 | Zbl 0721.58011

[21] N. Trudinger, Remarks concerning the conformal deformation of Riemannian structures on compact manifolds, Ann. Sc. Norm. Super. Pisa 22 (1968), 265-274 | Numdam | MR 240748 | Zbl 0159.23801

[22] H. Yamabe, On a deformation of Riemannian structures on compact manifolds, Osaka J. Math. 12 (1960), 21-37 | MR 125546 | Zbl 0096.37201

[23] S.T. Yau, Remarks on conformal transformations, J. Differential Geom. 8 (1973), 369-381 | MR 339007 | Zbl 0274.53047

[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 905611 | Zbl 0770.58005

[25] B. White, The space of minimal submanifolds for varying Riemannian metrics, Indiana Univ. Math. J. 40 (1991), 161-200 | MR 1101226 | Zbl 0742.58009