Examples of multiple solutions for the Yamabe problem on scalar curvature

Lionel Bérard Bergery (1, 2) , Guy Kass (1, 2)

1 IECN - Institut Élie Cartan de Nancy
2 Laboratoire de Mathématiques

In the conformal class of Riemannian metric on a compact connected manifold, there exists at least one metric with constant scalar curvature. In the case with positive scalar curvature, there may be many (non-homothtic) metrics with constant scalar curvature in a conformal class. R. Schoen gave a beautiful example of that phenomenon for a one-parameter family of metrics on $S^1 \times S^{n-1}$. In a preceding paper, we showed that Shoen's construction may be generalized on products $S^1 \times N$ (and other related examples). A unique (ordinary) differential equation, depending only on the dimenson, is the key to that construction. Here we give some more details on the solutions of that equation and their behavior on a one-parameter family.

Mots clés: Riemannian manifold; scalar curvature; conformal class; Yamabe problem

Domaines: Géométrie différentielle [math.DG]

Lionel Bérard Bergery, Guy Kass. Examples of multiple solutions for the Yamabe problem on scalar curvature. 2007. ⟨hal-00143495⟩