Abstract
This paper studies Sobolev type inequalities on Riemannian manifolds. We show that on a complete non-compact Riemannian manifold the constant in the Gagliardo–Nirenberg inequality cannot be smaller than the optimal one on the Euclidean space of the same dimension. We also show that a complete non-compact manifold with asymptotically non-negative Ricci curvature admitting some Gagliardo–Nirenberg inequality is not very far from the Euclidean space.
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