a Instituto de Matemática e Estatística, Universidade Federal de Goiás, 74001-900-Goiânia-GO, Brazil b Departamento de Matemática, Universidade de Brasilia, 70910-900-Brasilia-DF, Brazil
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.