Best constants in Sobolev inequalities

Let (M,g) be a smooth compact Riemannian N-manifold, with N greater than or equal to 2, and let p is an element of (1,N) be real, and H-1(p)(M) be the standard Sobolev space of order p. By the Sobolev embedding theorem, we have the inclusion H-1(p)(M) subset of L-p* (M), where p* = Np/(N - p) Classically, this leads to some Sobolev inequality (I-p(1)), and then to some Sobolev inequality (I-p(p)), where each term in (I-p(1)) is elevated to the power p. Long standing questions were to know if the optimal versions with respect to the first constant of (I-p(1)) and (I-p(p)) do hold. Such questions received an affirmative answer by Hebey-Vaugon for p = 2. We prove here that, for p > 2 with p(2) < N, the optimal version of (I-p(p)) is false if the scalar curvature of g is positive somewhere. In particular, there exist manifolds for which the optimal versions of (I-p(1)) is true, while the optimal version of (I-p(p)) is false. Among other results, we prove also that the assumption on the sign of the scalar curvature is minimal by showing that for any p is an element of (1, N), the optimal version of (I-p(p)) holds on flat tori. (C) Academie des Sciences/Elsevier, Paris.