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 $\varepsilon>0$, there exists $C_\varepsilon>0$ such that for any $u\in H_1^n(M)$, $\int_Me^u\mathrm{dv}_g\le C_\varepsilon\exp[(\mu_n+\varepsilon)\int_M\vert\nabla u\vert^n\mathrm{dv}_g+\frac{1}{\mathrm{vol}(M)}\int_Mu\mathrm{dv}_g]$, $\mu_n$ being the smallest constant possible such that the inequality remains true for any $u\in H_1^n(M)$. We call $\mu_n$ the ``first best constant''. We prove in this paper that it is possible to choose $\varepsilon=0$ and keep $C_\varepsilon$ 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.