Sharp Morrey-Sobolev inequalities on complete Riemannian Manifolds

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Two Morrey-Sobolev inequalities (with support-bound and L¹-bound, respectively) are investigated on complete Riemannian manifolds with their sharp constants in Rⁿ. We prove the following results in both cases: If (M,g) is a 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 (Rⁿ,e). If (M,g) has non-negative Ricci curvature, (M,g) supports the sharp Morrey-Sobolev inequalities if and only if (M,g) is isometric to (Rⁿ,e).