Sharp Morrey-Sobolev inequalities on complete Riemannian Manifolds

Abstract

Two Morrey-Sobolev inequalities (with support-bound and $L^1-$bound, respectively) are investigated on complete Riemannian manifolds with their sharp constants in ${\mathbb{R}}^n$. We prove the following results in both cases: $\bullet$ If $(M,g)$ is a {\it Cartan-Hadamard manifold} which verifies the $n-$dimensional Cartan-Hadamard conjecture, sharp Morrey-Sobolev inequalities hold on $(M,g)$. Moreover, extremals exist if and only if $(M,g)$ is isometric to the standard Euclidean space $({\mathbb{R}}^n,e)$. $\bullet$ If $(M,g)$ has {\it non-negative Ricci curvature}, $(M,g)$ supports the sharp Morrey-Sobolev inequalities if and only if $(M,g)$ is isometric to $({\mathbb{R}}^n,e)$.

15 pages, in press (Potential Analysis, 2014)