Sharp Sobolev-Poincare inequalities on compact Riemannian manifolds

Given (M;g) a smooth compact Riemannian n-manifold, n greater than or equal to 3, we return in this article to the study of the sharp Sobolev-Poincare type-inequality (0.1) parallel tou parallel to (2)(2*) less than or equal to K(n)(2)paralle l to delu parallel to (2)(2) + B parallel tou parallel to (2)(1) where 2(star) = 2n/(n - 2) is the critical Sobolev exponent, and K-n is the sharp Euclidean Sobolev constant. Druet, Hebey and Vaugon proved that (0.1 ) is true if n = 3, that(0.1) is true if n greater than or equal to 4 and t he sectional curvature of g is a nonpositive constant, or the Cartan-Hadama rd conjecture in dimension n is true and the sectional curvature of g is no npositive, but that (0.1) is false if n greater than or equal to 4 and the scalar curvature of g is positive somewhere. When (0.1) is true, we define B(g) as the smallest B in (0.1). The saturated form of (0.1) reads as (0.2) parallel tou parallel to (2)(2 star) less than or equal to K(n)(2)par allel to delu parallel to (2)(2) + B(g)parallel tou parallel to (2)(1). We assume in this article that n greater than or equal to 4, and complete t he study by Druet, Hebey and Vaugon of the sharp Sobolev-Poincare inequalit y (0.1). We prove that (0.1) is true, and that (0.2) possesses extremal fun ctions when the scalar curvature of g is negative. A fairly complete answer to the question of the validity of (0.1) under the assumption that the sca lar curvature is not necessarily negative, but only nonpositive, is also gi ven.