Optimal Transport: A Semi-Discrete Approach

For Riemannian manifolds with a measure (M, g, edvolg) we prove mean curvature and volume comparison results when the ∞-Bakry-Emery Ricci tensor is bounded from below and f is bounded or ∂rf is bounded from below, generalizing the classical ones (i.e. when f is constant). This leads to extensions of many theorems for Ricci curvature bounded below to the Bakry-Emery Ricci tensor. In particular, we give extensions of all of the major comparison theorems when f is bounded. Simple examples show the bound on f is necessary for these results.

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Comparison geometry for the Bakry-Emery Ricci tensor

Certain conditions for a Riemannian manifold to be isometric with a sphere:Dedicated to Professor Kentaro Yano on his fiftieth birthday

For Riemannian manifolds with a measure $(M,g, e^{-f} dvol_g)$ we prove mean curvature and volume comparison results when the $\infty$-Bakry-Emery Ricci tensor is bounded from below and $f$ is bounded or $\partial_r f$ is bounded from below, generalizing the classical ones (i.e. when $f$ is constant). This leads to extensions of many theorems for Ricci curvature bounded below to the Bakry-Emery Ricci tensor. In particular, we give extensions of all of the major comparison theorems when $f$ is bounded. Simple examples show the bound on $f$ is necessary for these results.

Comparison Geometry for the Bakry-Emery Ricci Tensor

The Bakry-Emery 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-Emery tensor. We show
that the Bakry-Emery tensor is nondecreasing under a Riemannian
submersion whose fiber transport preserves measures up to constants.
We give some
relations between the Bakry-Emery tensor and measured
Gromov-Hausdorff limits.

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Some geometric properties of the Bakry-Émery-Ricci tensor

We call a metric quasi-Einstein if the m -Bakry–Emery Ricci tensor is a constant multiple of the metric tensor. This is a generalization of Einstein metrics, it contains gradient Ricci solitons and is also closely related to the construction of the warped product Einstein metrics. We study properties of quasi-Einstein metrics and prove several rigidity results. We also give a splitting theorem for some Kahler quasi-Einstein metrics.

Rigidity of quasi-Einstein metrics ☆

In this paper we shall generalize the Bishop-Gromov relative volume comparison estimate to a situation where one only has an integral bound for the part of the Ricci curvature which lies below a given number. This will yield several compactness and pinching theorems.

Relative volume comparison with integral curvature bounds

We call a metric quasi-Einstein if the $m$-Bakry-Emery Ricci tensor is a constant multiple of the metric tensor. This is a generalization of Einstein metrics, which contains gradient Ricci solitons and is also closely related to the construction of the warped product Einstein metrics. We study properties of quasi-Einstein metrics and prove several rigidity results. We also give a splitting theorem for some K\"ahler quasi-Einstein metrics.

Guofang Wei

Papers

Comparison geometry for the Bakry-Emery Ricci tensor

Comparison Geometry for the Bakry-Emery Ricci Tensor

Rigidity of quasi-Einstein metrics ☆

Relative volume comparison with integral curvature bounds

Rigidity of Quasi-Einstein Metrics