Inégalité de Sobolev et volume asymptotique

Gilles Carron (1)

1 LMJL - Laboratoire de Mathématiques Jean Leray

A result of M. Ledoux is that a complete Riemannian manifold with non negative Ricci curvature satisfying the Euclidean Sobolev inequality is the Euclidean space. We present a shortcut of the proof. We also give a refinement of a result of B-L. Chen et X-P. Zhu about locally conformally flat manifolds with non negative Ricci curvature. Eventually, we discuss Ledoux's result when the hypothesis on the Ricci curvature is weakened on a hypothesis on the scalar curvature.

Mots clés: Sobolev inequality; volume growth; locally conformally flat

Domaines: Géométrie différentielle [math.DG]

Gilles Carron. Inégalité de Sobolev et volume asymptotique. Annales de la Faculté des Sciences de Toulouse. Mathématiques., 2012, 21 (1), pp.151-172. ⟨hal-00612419⟩