Sharp Sobolev-Poincaré inequalities on compact Riemannian manifolds |
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Authors: | Emmanuel Hebey |
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Institution: | Département de Mathématiques, Site de Saint-Martin, Université de Cergy-Pontoise, 2 avenue Adolphe Chauvin, 95302 Cergy-Pontoise cedex, France |
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Abstract: | Given a smooth compact Riemannian -manifold, , we return in this article to the study of the sharp Sobolev-Poincaré type inequality where is the critical Sobolev exponent, and is the sharp Euclidean Sobolev constant. Druet, Hebey and Vaugon proved that is true if , that is true if and the sectional curvature of is a nonpositive constant, or the Cartan-Hadamard conjecture in dimension is true and the sectional curvature of is nonpositive, but that is false if and the scalar curvature of is positive somewhere. When is true, we define as the smallest in . The saturated form of reads as We assume in this article that , and complete the study by Druet, Hebey and Vaugon of the sharp Sobolev-Poincaré inequality . We prove that is true, and that possesses extremal functions when the scalar curvature of is negative. A fairly complete answer to the question of the validity of under the assumption that the scalar curvature is not necessarily negative, but only nonpositive, is also given. |
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