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

Abstract:

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 $u\in H_1^n(M)$, $\int _Me^u\mathrm {dv}_g\le C_\varepsilon \exp [(\mu _n+\varepsilon )\int _M|\nabla u|^n\mathrm {dv}_g+\frac {1}{\mathrm {vol}(M)}\int _Mu\mathrm {dv}_g]$, $\mu _n$ 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.