Infinitely many solutions to the Yamabe problem on noncompact manifolds

Bettiol, Renato G.; Piccione, Paolo

Annales de l'Institut Fourier, Tome 68 (2018) no. 2, pp. 589-609.

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, ${𝕊}^m\times {ℝ}^d$, $m\ge 2$, $d\ge 1$, et ${𝕊}^m\times {ℍ}^d$, $2\le d<m$. Par conséquent, on obtient une infinité de solutions périodiques au problème de Yamabe singulier sur ${𝕊}^m\setminus {𝕊}^k$ pour tout $0\le k<(m-2)/2$, l’ensemble maximal pour laquelle la non-unicité est possible. Nous montrons également que tous les groupes de Bieberbach sur $\mathrm{Iso}\left({ℝ}^d\right)$ sont des périodes de branches de bifurcation de solutions de Yamabe sur ${𝕊}^m\times {ℝ}^d$, $m\ge 2$, $d\ge 1$.

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, ${𝕊}^m\times {ℝ}^d$, $m\ge 2$, $d\ge 1$, and ${𝕊}^m\times {ℍ}^d$, $2\le d<m$. As a consequence, we obtain infinitely many periodic solutions to the singular Yamabe problem on ${𝕊}^m\setminus {𝕊}^k$, for all $0\le k<(m-2)/2$, the maximal range where nonuniqueness is possible. We also show that all Bieberbach groups in $\mathrm{Iso}\left({ℝ}^d\right)$ are periods of bifurcating branches of solutions to the Yamabe problem on ${𝕊}^m\times {ℝ}^d$, $m\ge 2$, $d\ge 1$.

Classification : 53A30, 53C21, 35J60, 58J55, 58E11, 58E15
Keywords: Yamabe problem, singular Yamabe problem, Constant scalar curvature, nonuniqueness of solutions, Aubin’s inequality, bifurcation
Mot clés : Problème de Yamabe, problème de Yamabe singulier, courbure scalaire constante, non-unicité des solutions, inégalité de Aubin, bifurcation

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