Sharp Morrey-Sobolev inequalities on complete Riemannian Manifolds

TL;DR

This paper establishes sharp Morrey-Sobolev inequalities on complete Riemannian manifolds under certain curvature conditions, characterizing when extremals exist and identifying Euclidean space as the unique case for equality.

Contribution

It proves sharp Morrey-Sobolev inequalities on Cartan-Hadamard and non-negatively curved manifolds, characterizing extremals and the Euclidean space as the only extremal case.

Findings

Sharp inequalities hold on Cartan-Hadamard manifolds satisfying the conjecture.

Extremals exist only when the manifold is isometric to Euclidean space.

Manifolds with non-negative Ricci curvature support inequalities only if they are Euclidean.

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 $(\mathbb R^n,e)$.