Infinitely many solutions to the Yamabe problem on noncompact manifolds
Annales de l'Institut Fourier, Volume 68 (2018) no. 2, p. 589-609
We establish the existence of infinitely many complete metrics with constant scalar curvature in conformal classes of certain noncompact product manifolds. These include products of closed manifolds with constant positive scalar curvature and simply-connected symmetric spaces of noncompact or Euclidean type; in particular, 𝕊 m × d , m2, d1, and 𝕊 m × d , 2d<m. As a consequence, we obtain infinitely many periodic solutions to the singular Yamabe problem on 𝕊 m 𝕊 k , for all 0k<(m-2)/2, the maximal range where nonuniqueness is possible. We also show that all Bieberbach groups in Iso( d ) are periods of bifurcating branches of solutions to the Yamabe problem on 𝕊 m × d , m2, d1.
On établit l’existence d’une infinité de métriques complètes à courbure scalaire constante dans une classe conforme prescrite sur des variétés produit non-compactes. Celles-ci incluent produits de variétés fermés à courbure scalaire constante et des espaces symétriques simplement connexes de type non-compact ou Euclidien. En particulier, 𝕊 m × d , m2, d1, et 𝕊 m × d , 2d<m. Par conséquent, on obtient une infinité de solutions périodiques au problème de Yamabe singulier sur 𝕊 m 𝕊 k pour tout 0k<(m-2)/2, l’ensemble maximal pour laquelle la non-unicité est possible. Nous montrons également que tous les groupes de Bieberbach sur Iso( d ) sont des périodes de branches de bifurcation de solutions de Yamabe sur 𝕊 m × d , m2, d1.
Received : 2016-10-22
Accepted : 2017-09-19
Published online : 2018-04-18
DOI : https://doi.org/10.5802/aif.3172
Classification:  53A30,  53C21,  35J60,  58J55,  58E11,  58E15
Keywords: Yamabe problem, singular Yamabe problem, Constant scalar curvature, nonuniqueness of solutions, Aubin’s inequality, bifurcation
@article{AIF_2018__68_2_589_0,
     author = {Bettiol, Renato G. and Piccione, Paolo},
     title = {Infinitely many solutions to the Yamabe problem on noncompact manifolds},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {68},
     number = {2},
     year = {2018},
     pages = {589-609},
     doi = {10.5802/aif.3172},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2018__68_2_589_0}
}
Infinitely many solutions to the Yamabe problem on noncompact manifolds. Annales de l'Institut Fourier, Volume 68 (2018) no. 2, pp. 589-609. doi : 10.5802/aif.3172. https://aif.centre-mersenne.org/item/AIF_2018__68_2_589_0/

[1] Akutagawa, Kazuo; Florit, Luis A.; Petean, Jimmy On Yamabe constants of Riemannian products, Commun. Anal. Geom., Tome 15 (2007) no. 5, pp. 947-969 http://projecteuclid.org/euclid.cag/1210944225 | Article | MR 2403191 (2009i:53030) | Zbl 1147.53032

[2] Akutagawa, Kazuo; Neves, André 3-manifolds with Yamabe invariant greater than that of 3 , J. Differ. Geom., Tome 75 (2007) no. 3, pp. 359-386 http://projecteuclid.org/euclid.jdg/1175266277 | Article | MR 2301449 | Zbl 1119.53027

[3] Aubin, Thierry Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire, J. Math. Pures Appl., Tome 55 (1976) no. 3, pp. 269-296 | MR 0431287 (55 #4288) | Zbl 0336.53033

[4] Aubin, Thierry The scalar curvature, Differential geometry and relativity, Reidel (Mathematical Physics and Applied Mathematics) Tome 3 (1976), pp. 5-18 | MR 0433500 | Zbl 0345.53029

[5] Aubin, Thierry Some nonlinear problems in Riemannian geometry, Springer, Springer Monographs in Mathematics (1998), xviii+395 pages | Article | MR 1636569 (99i:58001) | Zbl 0896.53003

[6] Aviles, Patricio; Mcowen, Robert C. Conformal deformation to constant negative scalar curvature on noncompact Riemannian manifolds, J. Differ. Geom., Tome 27 (1988) no. 2, pp. 225-239 http://projecteuclid.org/euclid.jdg/1214441781 | Article | MR 925121 (89b:58225) | Zbl 0648.53021

[7] Berger, Marcel; Gauduchon, Paul; Mazet, Edmond Le spectre d’une variété riemannienne, Springer, Lecture Notes in Mathematics, Tome 194 (1971), vii+251 pages | MR 0282313 | Zbl 0223.53034

[8] Berti, Massimiliano; Malchiodi, Andrea Non-compactness and multiplicity results for the Yamabe problem on S n , J. Funct. Anal., Tome 180 (2001) no. 1, pp. 210-241 | Article | MR 1814428 (2002b:53049) | Zbl 0979.53038

[9] Bettiol, Renato G.; Derdzinski, Andrzej; Piccione, Paolo Teichmüller theory and collapse of flat manifolds (2017) (to appear in Ann. Mat. Pura Appl., http://arxiv.org/abs/1705.08431)

[10] Bettiol, Renato G.; Piccione, Paolo Bifurcation and local rigidity of homogeneous solutions to the Yamabe problem on spheres, Calc. Var. Partial Differ. Equ., Tome 47 (2013) no. 3-4, pp. 789-807 | Article | MR 3070564 | Zbl 1272.53042

[11] Bettiol, Renato G.; Piccione, Paolo Multiplicity of solutions to the Yamabe problem on collapsing Riemannian submersions, Pac. J. Math., Tome 266 (2013) no. 1, pp. 1-21 | Article | MR 3105774 | Zbl 1287.53030

[12] Bettiol, Renato G.; Piccione, Paolo; Santoro, Bianca Bifurcation of periodic solutions to the singular Yamabe problem on spheres, J. Differ. Geom., Tome 103 (2016) no. 2, pp. 191-205 http://projecteuclid.org/euclid.jdg/1463404117 | Article | MR 3504948 | Zbl 1348.53044

[13] Borel, Armand Compact Clifford-Klein forms of symmetric spaces, Topology, Tome 2 (1963), pp. 111-122 | Article | MR 0146301 (26 #3823) | Zbl 0116.38603

[14] Brandolini, Luca; Rigoli, Marco; Setti, Alberto G. Positive solutions of Yamabe type equations on complete manifolds and applications, J. Funct. Anal., Tome 160 (1998) no. 1, pp. 176-222 | Article | MR 1658696 (2000a:35051) | Zbl 0923.58049

[15] Buser, Peter A geometric proof of Bieberbach’s theorems on crystallographic groups, Enseign. Math., Tome 31 (1985) no. 1-2, pp. 137-145 | MR 798909 | Zbl 0582.20033

[16] Caffarelli, Luis A.; Gidas, Basilis; Spruck, Joel Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Commun. Pure Appl. Math., Tome 42 (1989) no. 3, pp. 271-297 | Article | MR 982351 (90c:35075) | Zbl 0702.35085

[17] Carlotto, Alessandro; Chodosh, Otis; Rubinstein, Yanir A. Slowly converging Yamabe flows, Geom. Topol., Tome 19 (2015) no. 3, pp. 1523-1568 | Article | MR 3352243 | Zbl 1326.53089

[18] Charlap, Leonard S. Bieberbach groups and flat manifolds, Springer, Universitext (1986), xiv+242 pages | Article | MR 862114 | Zbl 0608.53001

[19] Cheng, Shiu Yuen Eigenvalue comparison theorems and its geometric applications, Math. Z., Tome 143 (1975) no. 3, pp. 289-297 | Article | MR 0378001 | Zbl 0329.53035

[20] Deligne, Pierre Extensions centrales non résiduellement finies de groupes arithmétiques, C. R. Acad. Sci., Paris, Sér. A, Tome 287 (1978) no. 4, p. A203-A208 | MR 507760 (80g:20056) | Zbl 0416.20042

[21] Druţu, Cornelia; Sapir, Mark Non-linear residually finite groups, J. Algebra, Tome 284 (2005) no. 1, pp. 174-178 | Article | MR 2115010 (2005i:20045) | Zbl 1076.20021

[22] Ferrand, Jacqueline The action of conformal transformations on a Riemannian manifold, Math. Ann., Tome 304 (1996) no. 2, pp. 277-291 | Article | MR 1371767 (97c:53044) | Zbl 0866.53027

[23] Grosse, Nadine The Yamabe equation on manifolds of bounded geometry, Commun. Anal. Geom., Tome 21 (2013) no. 5, pp. 957-978 | Article | MR 3152969 | Zbl 1296.53076

[24] Hebey, Emmanuel; Vaugon, Michel Meilleures constantes dans le théorème d’inclusion de Sobolev et multiplicité pour les problèmes de Nirenberg et Yamabe, Indiana Univ. Math. J., Tome 41 (1992) no. 2, pp. 377-407 | Article | MR 1183349 (93m:58121) | Zbl 0764.53029

[25] Henry, Guillermo; Petean, Jimmy Isoparametric hypersurfaces and metrics of constant scalar curvature, Asian J. Math., Tome 18 (2014) no. 1, pp. 53-67 | Article | MR 3215339 | Zbl 1292.53041

[26] Henry, Guillermo; Petean, Jimmy On Yamabe constants of products with hyperbolic spaces, J. Geom. Anal., Tome 25 (2015) no. 2, pp. 1387-1400 | Article | MR 3319976 | Zbl 1318.53031

[27] Hiss, Gerhard; Szczepański, Andrzej On torsion free crystallographic groups, J. Pure Appl. Algebra, Tome 74 (1991) no. 1, pp. 39-56 | Article | MR 1129128 (92i:20053) | Zbl 0760.20015

[28] Jin, Zhi Ren A counterexample to the Yamabe problem for complete noncompact manifolds, Partial differential equations (Tianjin, 1986), Springer (Lecture Notes in Mathematics) Tome 1306 (1988), pp. 93-101 | Article | MR 1032773 (91a:53065) | Zbl 0648.53022

[29] Kobayashi, Osamu Scalar curvature of a metric with unit volume, Math. Ann., Tome 279 (1987) no. 2, pp. 253-265 | Article | MR 919505 (89a:53048) | Zbl 0611.53037

[30] Lee, John M.; Parker, Thomas H. The Yamabe problem, Bull. Am. Math. Soc., Tome 17 (1987) no. 1, pp. 37-91 | Article | MR 888880 (88f:53001) | Zbl 0633.53062

[31] De Lima, Levi L.; Piccione, Paolo; Zedda, Michela On bifurcation of solutions of the Yamabe problem in product manifolds, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Tome 29 (2012) no. 2, pp. 261-277 | Article | MR 2901197 | Zbl 1239.58005

[32] Long, Darren; Reid, Alan W. Surface subgroups and subgroup separability in 3-manifold topology (Rio de Janeiro, 2005), Instituto Nacional de Matemática Pura e Aplicada (IMPA), Publicações Matemáticas do IMPA (2005), 53 pages | MR 2164951 (2006f:57018) | Zbl 1074.57010

[33] Mazzeo, Rafe Regularity for the Singular Yamabe Problem, Indiana Univ. Math. J., Tome 40 (1991) no. 4, pp. 1277-1299 | Article | MR 1142715 (92k:53071) | Zbl 0770.53032

[34] Mazzeo, Rafe; Smale, Nathan Conformally flat metrics of constant positive scalar curvature on subdomains of the sphere, J. Differ. Geom., Tome 34 (1991) no. 3, pp. 581-621 http://projecteuclid.org/euclid.jdg/1214447536 | Article | MR 1139641 (92i:53034) | Zbl 0759.53029

[35] Petean, Jimmy; Ruiz, Juan Miguel On the Yamabe constants of S 2 × 3 and S 3 × 2 , Differ. Geom. Appl., Tome 31 (2013) no. 2, pp. 308-319 | Article | MR 3032651 | Zbl 1280.53042

[36] Pollack, Daniel Nonuniqueness and high energy solutions for a conformally invariant scalar equation, Commun. Anal. Geom., Tome 1 (1993) no. 3-4, pp. 347-414 | Article | MR 1266473 (94m:58051) | Zbl 0848.58011

[37] Ramírez-Ospina, Héctor Fabián Multiplicity of constant scalar curvature metrics in T k ×M, Nonlinear Anal., Tome 109 (2014), pp. 103-112 | Article | MR 3247296 | Zbl 1296.58006

[38] Ratcliffe, John G. Foundations of hyperbolic manifolds, Springer, Graduate Texts in Mathematics, Tome 149 (2006), xii+779 pages | MR 2249478 (2007d:57029) | Zbl 1106.51009

[39] Schoen, Richard Conformal deformation of a Riemannian metric to constant scalar curvature, J. Differ. Geom., Tome 20 (1984) no. 2, pp. 479-495 http://projecteuclid.org/euclid.jdg/1214439291 | Article | MR 788292 (86i:58137) | Zbl 0576.53028

[40] Schoen, Richard Variational theory for the total scalar curvature functional for Riemannian metrics and related topics, Topics in calculus of variations (Montecatini Terme, 1987), Springer (Lecture Notes in Mathematics) Tome 1365 (1989), pp. 120-154 | Article | MR 994021 (90g:58023) | Zbl 0702.49038

[41] Schoen, Richard On the conformal and CR automorphism groups, Geom. Funct. Anal., Tome 5 (1995) no. 2, pp. 464-481 | Article | MR 1334876 (96h:53047) | Zbl 0835.53015

[42] Szczepański, Andrzej Geometry of crystallographic groups, World Scientific, Algebra and Discrete Mathematics, Tome 4 (2012), xii+195 pages | Article | MR 2978307 | Zbl 1260.20070

[43] Trudinger, Neil S. Remarks concerning the conformal deformation of Riemannian structures on compact manifolds, Ann. Sc. Norm. Super. Pisa, Tome 22 (1968), pp. 265-274 | MR 0240748 (39 #2093) | Zbl 0159.23801

[44] Wolf, Joseph A. Local and global equivalence for flat affine manifolds with parallel geometric structures, Geom. Dedicata, Tome 2 (1973), pp. 127-132 | Article | MR 0322748 | Zbl 0279.53039

[45] Yamabe, Hidehiko On a deformation of Riemannian structures on compact manifolds, Osaka Math. J., Tome 12 (1960), pp. 21-37 | MR 0125546 (23 #A2847) | Zbl 0096.37201