On the geometric dependence of Riemannian Sobolev best constants (Q1035028)

Let \(M\) be a smooth compact manifold of dimension \(n\geq 2\). Denote by \(\mathcal{M}_2\) the space of smooth Riemannian metrics on \(M\) endowed with the \(\mathcal{C}^2\)-topology and by \(\mathcal{M}_\infty\) the space of smooth Riemannian metrics on \(M\) endowed with the usual Fréchet topology. The authors prove the following results. Theorem 1.1. Let \(M\) be a smooth compact manifold of dimension \(n\). If \(n\geq 4\), then the map \(f\in \mathcal{M}_2\to B_0(2,g)\) is continuous. Moreover, the \(\mathcal{C}^2\)-topology is sharp. Theorem 1.2. Let \(M\) be a smooth compact manifold of dimension \(n\). If \(n\geq 2\) and \(1\leq p\leq \min\{2, \sqrt{n}\}\) then the map \(g\in \mathcal{M}_2\to B_0(p,g)\) is continuous. Some immediate consequences are presented.