A sharp Sobolev inequality on Riemannian manifolds

Let (M, g) be a smooth compact Riemannian manifold without boundary of dimension n ≥ 6. We prove that ‖u‖ L2 ∗ (M,g) ≤ K 2 ∫ M { |∇gu| 2 + c(n)Rgu 2 } dvg + A‖u‖ 2 L2n/(n+2)(M,g), for all u ∈ H(M), where 2 = 2n/(n − 2), c(n) = (n − 2)/[4(n − 1)], Rg is the scalar curvature, K −1 = inf ‖∇u‖L2(Rn)‖u‖ −1 L2n/(n−2)(Rn) and A > 0 is a constant depending on (M, g) only. The inequality is sharp in the sense that on any (M, g), K can not be replaced by any smaller number and Rg can not be replaced by any continuous function which is smaller than Rg at some point. If (M, g) is not locally conformally flat, the exponent 2n/(n + 2) can not be replaced by any smaller number. If (M, g) is locally conformally flat, a stronger inequality, with 2n/(n + 2) replaced by 1, holds in all dimensions n ≥ 3. key words: sharp Sobolev inequality, critical exponent, Yamabe problem MSC 2000 subject classification: 35J60, 58E35 Partially supported by National Science Foundation Grant DMS-0100819 and a Rutgers University Research Council Grant. Partially supported by CNR Fellowship 203.01.69 (19/01/98) and by PRIN 2000 “Variational Methods and Nonlinear Differential Equations”.