Sharp Sobolev inequalities of second order

Let (M, g) be a smooth compact Riemannian manifold of dimension n≥5, and ₂ ² (M) be the Sobolev space consisting of functions in L²(M) whose derivatives up to the order two are also in L²(M). Thanks to the Sobolev embedding theorem, there exist positive constants A and B such that for any U ∈ H ₂ ² (M), where 2^#=2n/(n−4) is critical, and is the usual norm on the Sobolev space H ₁ ² (M) consisting of functions in L²(M) whose derivatives of order one are also in L²(M). The sharp constant A in this inequality is K ₀ ² where K₀, an explicit constant depending only on n, is the sharp constant for the Euclidean Sobolev inequality . We prove in this article that for any compact Riemannian manifold, A=K ₀ ² is attained in the above inequality.