Best constants in Sobolev inequalities for compact manifolds of nonpositive curvature

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Let ( M,g) be a smooth compact Riemannian n-manifold, n ≥ 2, let p(1, n) real, and let H 1 p ( M) be the standard Sobolev space of order p. By the Sobolev embedding theorem, H 1 p ( M) ⊂ 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, and on what concerns (I p 1), by Aubin for two-dimensional manifolds and for manifolds of constant sectional curvature. Recently, Druet proved that for p > 2, and p 2 < n, the optimal version of (I p p ) is false if the scalar curvature of g is positive somewhere, while for p > 1, the optimal version of (I p p ) does hold on flat torii and compact hyperbolic spaces. We prove here that the optimal version of (I p p ), p > 1, does hold for compact manifolds of nonpositive sectional curvature in any dimension where the Cartan-Hadamard conjecture is true. In particular, since the Cartan-Hadmard conjecture is true in dimensions 2, 3, and 4, the optimal version of (I p p ) does hold on any compact manifold of nonpositive sectional curvature of dimension 2, 3, or 4.