Sharp Sobolev-Poincaré inequalities on compact Riemannian manifolds

Abstract

Given (M,g) a smooth compact Riemannian n-manifold, n > 3, we return in this article to the study of the sharp Sobolev-Poincare type inequality (0.1) ∥u∥ 2 2* ≤ K 2 n ∥Δ∥u∥ 2 2 + B∥u∥ 2 1 where 2 * = 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 > 4 and the sectional curvature of g is a nonpositive constant, or the Cartan-Hadamard conjecture in dimension n is true and the sectional curvature of g is nonpositive, but that (0.1) is false if n > 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) ∥u∥ 2 2* ≤ K 2 n ∥Δu∥ 2 2 + B(g)∥u∥ 2 1 . We assume in this article that n > 4, and complete the study by Druet, Hebey and Vaugon of the sharp Sobolev-Poincare inequality (0.1). We prove that (0.1) is true, and that (0.2) possesses extremal functions 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 scalar curvature is not necessarily negative, but only nonpositive, is also given.