Optimal constants in the exceptional case of Sobolev inequalities on Riemannian manifolds

Let ( M, g) be a Riemannian compact n-manifold. We know that for any epsilon > 0, there exists C-epsilon > 0 such that for any u epsilon H-1(n) (M), integral(M)e(u)dv (g) <= C-epsilon exp[(mu(n)+ epsilon) integral(M) vertical bar del u vertical bar(n)dv(g) + 1/vol (M) integral(M) udv (g)], mu(n) being the smallest constant possible such that the inequality remains true for any u epsilon H-1(n) (M). We call mu(n) the "first best constant". We prove in this paper that it is possible to choose epsilon = 0 and keep C-epsilon a finite constant. In other words we prove the existence of a "second best constant" in the exceptional case of Sobolev inequalities on compact Riemannian manifolds.