Weighted Poincaré inequality and rigidity of complete manifolds

By Peter Li, Jiaping Wang. 2006

146 citations recorded by the source corpus.

Abstract We prove structure theorems for complete manifolds satisfying both the Ricci curvature lower bound and the weighted Poincare 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 Poincare inequality, and yields a criterion of parabolicity of connected components at infinity in terms of the weight function.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/b2781465edde3e5c07b8acfc686ff7b7a5189183

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Hardy-Poincaré, Rellich and uncertainty principle inequalities on Riemannian manifolds (2011, unknown): “[6], [8], [13], [3], [17], [22], [16], [18], and the references therein).”
Analysis of weighted Laplacian and applications to Ricci solitons (2011, background): “Li and the second author in [16, 17, 18, 19] and our generalization in [20] to the case of weighted Laplacian with Ricf ≥ 0, we study the structure of manifolds on which λ1 (∆f ) achieves its maximal value in the preceding estimate.”
On Gradient Ricci Solitons (2009, background): “Weighted Poincaré inequalities are known to be equivalent to the manifold being nonparabolic [21].”
OPTIMAL HARDY WEIGHT FOR SECOND-ORDER ELLIPTIC OPERATOR: AN ANSWER TO A PROBLEM OF AGMON (2012, background): “Carron’s results have been later rediscovered by Li and Wang [35], where the authors give also applications to structure theorems for complete manifolds.”
Sharp uncertainty principles on Riemannian manifolds: the influence of curvature (2013, background): “2 can be included into the best constant program initiated by Aubin [4], and studied by Ledoux [30], Cheeger and Colding [12], Druet, Hebey and Vaugon [16], do Carmo and Xia [15], Minerbe [32], Li and Wang [31], etc.”
Sharp logarithmic Sobolev inequalities on gradient solitons and applications (2008, background): “The weighted Poincaré inequality and its interplay with the geometry has recently been studied in [23].”
Improved Hardy and Rellich inequalities on Riemannian manifolds (2007, background): “Recently, Li and Wang [17] proved that, among other results, existence of a weighted Hardy type inequality is equivalent to nonparabolicity.”
Prime ends for domains in metric spaces (2012, unknown): “See [23], [24], [28], [29], [38], [39], [40], [43], [52], [53] and [63] for some applications of parabolic ends.”
Hardy inequalities on Riemannian manifolds and applications (2012, methodology): “Notice that one of the implications of this characterization for p = 2 is the result proved in [40].”