Volume comparison theorems without Jacobi fields

By D. Bakry, Zhongmin Qian. 2005

107 citations recorded by the source corpus.

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Citations in the corpus

Beyond traditional Curvature-Dimension I: new model spaces for isoperimetric and concentration inequalities in negative dimension (2014, background): “…of Cordero-Erausquin–McCann–Schmuckenschl¨ager [18,19] involving distortion coeﬃcients, and as already described in Remark 6.2, a reduction by Klartag of the Poincar´e inequality to the one-dimensional case when N 1, extending a previous result by Bakry and Qian [4] for the case N ∈ [ n, ∞ ].”
Li-Yau-Hamilton estimates and Bakry-Emery Ricci curvature (2014, background): “Bakry-Qian’s comparison theorem If Ric V ≥ K for some constant K, then the elliptic operator ∆V satisfies the CD(K,n) condition in the sense of Bakry [3], see also [6, 24].”
On the classification of warped product Einstein metrics (2010, methodology): “On one hand we know from [BQ] that volume growth with respect to this volume form is a power function of degree ≤ n+m.”
Perelman’s entropy formula for the Witten Laplacian on Riemannian manifolds via Bakry–Emery Ricci curvature (2012, background): “By the generalized Laplacian comparison theorem [ 5 ,23,36,37], under the condition Ricm,n(L) ≥− (m − 1)K ,w e have”
Comparison geometry for the Bakry-Emery Ricci tensor (2007, background): “This is a special case of an estimate in [6], but we have written the result in more Riemannian geometry friendly language.”
Mass Transport and Variants of the Logarithmic Sobolev Inequality (2007, methodology): “[10]): let x0 ∈M , then D = M \Cut(x0) is an x0- star-shaped domain.”
Liouville theorems for symmetric diffusion operators on complete Riemannian manifolds (2005, methodology): “Clearly, the assumption required in our Theorem 1.5 is weaker than Qian’s conditi L = − ∇φ · ∇.”
Poincaré and Brunn-Minkowski inequalities on the boundary of weighted Riemannian manifolds (2017, background): “, also [5, 6, 20, 21, 38, 51, 57, 64, 77, 78] and the references therein), especially since Perelman’s work on the Poincaré Conjecture [62], and the extension of the Curvature-Dimension condition to the metric-measure space setting by Lott-Sturm-Villani [46, 72].”