Infinitely many solutions to the Yamabe problem on noncompact manifolds
Annales de l'Institut Fourier, Volume 68 (2018) no. 2, pp. 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:
Accepted:
Published online:
DOI: 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
License: CC-BY-ND 4.0
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Bettiol, Renato G.; Piccione, Paolo. 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/articles/10.5802/aif.3172/

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