Abstract: | 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. |