Some geometric properties of the Bakry-Émery-Ricci tensor

By J. Lott. 2002

330 citations recorded by the source corpus.

Abstract The Bakry-Émery tensor gives an analog of the Ricci tensor for a Riemannian manifold with a smooth measure. We show that some of the topological consequences of having a positive or nonnegative Ricci tensor are also valid for the Bakry-Émery tensor. We show that the Bakry-Émery tensor is nondecreasing under a Riemannian submersion whose fiber transport preserves measures up to constants. We give some relations between the Bakry-Émery tensor and measured Gromov-Hausdorff limits.

Subfields: Applied Mathematics, Astronomy and Astrophysics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/215b1ab33719d3fae31fe23b24e642baa95d585c

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Citations in the corpus

Heat flow on Finsler manifolds (2008, background): “quality or Γ2-inequality in the sense of Bakry-Emery´ Γ2(v,v) ≥ 1 N (∆Zv)2 +K·Γ(v,v) (5.3) for all smooth functions von Mz. Here Γ(v,w) = Dv(∇Zw) and Γ2(v,v) = 1 2 ∆ZΓ(v,v) −Γ(∆Zv,v), see [BE], [Qi], [Lo]. The remarkable observation of Bakry and Qian [BQ] is the ‘self-improving property’ of (5.3) sa”
(K, N)-Convexity and the Curvature-Dimension Condition for Negative N (2013, neutral): “The parameter N had been usually chosen from [n,∞], and the bound RicN (v) ≥ K |v|2 is known to imply many analytic and geometric consequences corresponding to Ric ≥ K as well as dim ≤ N (see [21,37]).”
A remark on compact Ricci solitons (2008, background): “Topological implications of bounds on the Bakry-Emery tensor on closed manifolds have been discussed by Lott [ 15, Theorem 1 ], from where it follows that the fundamental group of M is finite for any compact soliton.”
The Bakry–Emery Ricci tensor and its applications to some compactness theorems (2012, methodology): “1 Introduction The purpose of this article is to generalize the well-known Myers compactness theorem [5] by using the Bakry–Emery Ricci curvature (see [1,4] and [6]) on a complete and connected Riemannian manifold (M, g) of dimension n.”
Li-Yau-Hamilton estimates and Bakry-Emery Ricci curvature (2014, background): “By the Bishop–Gromov volume comparison theorem for ∆f (see [28], or [37, Theorem 4.”
Optimal transport and Perelman’s reduced volume (2008, result): “This is in analogy to what works in defining Ricci tensors for smooth metric-measure spaces [9].”
Beyond traditional Curvature-Dimension I: new model spaces for isoperimetric and concentration inequalities in negative dimension (2014, methodology): “The generalized Ricci tensor (1.1) was introduced with N = ∞ by Lichnerowicz [36, 37] and in general by Bakry [1] (cf. Lott [38]).”
Sharp Isoperimetric Inequalities and Model Spaces for Curvature-Dimension-Diameter Condition (2011, neutral): “2) was noted in [52]), and has been extensively studied and used in recent years (see e.”
Ricci curvature for metric-measure spaces via optimal transport (2004, methodology): “Finally, we use displacement convexity to give a “synthetic” proof of a part of the Ricci O’Neill theorem from [30].”
On quotients of spaces with Ricci curvature bounded below (2017, background): “In [52] Lott calls this condition the fiber transport preserves the fiberwise measure.”
Comparison geometry for the Bakry-Emery Ricci tensor (2007, background): “D G ] 1 4 Se p 20 07”
A Liouville-Type Theorem for Smooth Metric Measure Spaces (2010, background): “See, for example [4, 7, 9, 12, 8] or [11] and the references therein.”