Weighted Poincaré inequality and rigidity of complete manifolds

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Abstract

We prove structure theorems for complete manifolds satisfying both the Ricci curvature lower bound and the weighted Poincaré inequality. In the process, a sharp decay estimate for the minimal positive Green's function is obtained. This estimate only depends on the weight function of the Poincaré inequality, and yields a criterion of parabolicity of connected components at infinity in terms of the weight function.

Résumé

Nous prouvons des théorèmes de structure pour des variétés complètes telles que la courbure de Ricci soit minorée, et satisfaisant l'inégalité de Poincaré à poids. Nous obtenons une estimation optimale de la décroissance de la fonction de Green positive et minimale. Cette estimation, qui dépend seulement du poids de la fonction dans l'inégalité de Poincaré, produit un critère de parabolicité de composantes connexes à l'infini utilisant le poids de la fonction.

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    1

    The first author was partially supported by NSF Grant DMS-0503735.

    2

    The second author was partially supported by NSF Grant DMS-0404817.

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