# Metric measure spaces with Riemannian Ricci curvature bounded from below

@article{Ambrosio2014MetricMS,
title={Metric measure spaces with Riemannian Ricci curvature bounded from below},
author={Luigi Ambrosio and Nicola Gigli and Giuseppe Savar'e},
journal={Duke Mathematical Journal},
year={2014},
volume={163},
pages={1405-1490}
}
• Published 1 September 2011
• Mathematics
• Duke Mathematical Journal
In this paper we introduce a synthetic notion of Riemannian Ricci bounds from below for metric measure spaces (X,d,m) which is stable under measured Gromov-Hausdorff convergence and rules out Finsler geometries. It can be given in terms of an enforcement of the Lott, Sturm and Villani geodesic convexity condition for the entropy coupled with the linearity of the heat flow. Besides stability, it enjoys the same tensorization, global-to-local and local-to-global properties. In these spaces, that…
489 Citations

### Riemannian Ricci curvature lower bounds in metric measure spaces with -finite measure

• Mathematics
• 2015
In prior work (4) of the first two authors with Savare, a new Riemannian notion of lower bound for Ricci curvature in the class of metric measure spaces (X,d,m) was introduced, and the corresponding

### Diffusion, Optimal Transport and Ricci Curvature for Metric Measure Space

• Mathematics
• 2017
Lower Ricci curvature bounds play a crucial role in several deep geometric and functional inequalities in Riemannian geometry and diffusion processes. Bakry–Émery [8] introduced an elegant and

### Heat Flow and Calculus on Metric Measure Spaces with Ricci Curvature Bounded Below—The Compact Case

• Mathematics
• 2012
We provide a quick overview of various calculus tools and of the main results concerning the heat flow on compact metric measure spaces, with applications to spaces with lower Ricci curvature bounds.

### Bakry-Émery curvature-dimension condition and Riemannian Ricci curvature bounds

• Mathematics
• 2015
The aim of the present paper is to bridge the gap between the Bakry–Emery and the Lott–Sturm–Villani approaches to provide synthetic and abstract notions of lower Ricci curvature bounds. We start

### Remarks about Synthetic Upper Ricci Bounds for Metric Measure Spaces

We discuss various characterizations of synthetic upper Ricci bounds for metric measure spaces in terms of heat flow, entropy and optimal transport. In particular, we present a characterization in

### The isoperimetric problem on Riemannian manifolds via Gromov-Hausdorff asymptotic analysis

• Mathematics
Communications in Contemporary Mathematics
• 2022
In this paper we prove the existence of isoperimetric regions of any volume in Riemannian manifolds with Ricci bounded below assuming Gromov–Hausdorﬀ asymptoticity to the suitable simply connected

## References

SHOWING 1-10 OF 78 REFERENCES

### Riemannian Ricci curvature lower bounds in metric measure spaces with -finite measure

• Mathematics
• 2015
In prior work (4) of the first two authors with Savare, a new Riemannian notion of lower bound for Ricci curvature in the class of metric measure spaces (X,d,m) was introduced, and the corresponding

### Heat Flow and Calculus on Metric Measure Spaces with Ricci Curvature Bounded Below—The Compact Case

• Mathematics
• 2012
We provide a quick overview of various calculus tools and of the main results concerning the heat flow on compact metric measure spaces, with applications to spaces with lower Ricci curvature bounds.

### Ricci curvature for metric-measure spaces via optimal transport

• Mathematics
• 2004
We dene a notion of a measured length space X having nonnegative N-Ricci curvature, for N 2 [1;1), or having1-Ricci curvature bounded below byK, forK2 R. The denitions are in terms of the

### Bakry-Émery curvature-dimension condition and Riemannian Ricci curvature bounds

• Mathematics
• 2015
The aim of the present paper is to bridge the gap between the Bakry–Emery and the Lott–Sturm–Villani approaches to provide synthetic and abstract notions of lower Ricci curvature bounds. We start

### On the differential structure of metric measure spaces and applications

The main goals of this paper are: i) To develop an abstract differential calculus on metric measure spaces by investigating the duality relations between differentials and gradients of Sobolev

### Lower bounds on Ricci curvature and the almost rigidity of warped products

• Mathematics
• 1996
The basic rigidity theorems for manifolds of nonnegative or positive Ricci curvature are the "volume cone implies metric cone" theorem, the maximal diameter theorem, [Cg], and the splitting theorem,

### On the structure of spaces with Ricci curvature bounded below. II

• Mathematics
• 2000
In this paper and in we study the structure of spaces Y which are pointed Gromov Hausdor limits of sequences f M i pi g of complete connected Riemannian manifolds whose Ricci curvatures have a de

### On the equivalence of the entropic curvature-dimension condition and Bochner’s inequality on metric measure spaces

• Mathematics
• 2013
We prove the equivalence of the curvature-dimension bounds of Lott–Sturm–Villani (via entropy and optimal transport) and of Bakry–Émery (via energy and $$\Gamma _2$$Γ2-calculus) in complete

### Eulerian Calculus for the Displacement Convexity in the Wasserstein Distance

• Mathematics
SIAM J. Math. Anal.
• 2008
A new proof of the (strong) displacement convexity of a class of integral functionals defined on a compact Riemannian manifold satisfying a lower Ricci curvature bound is given.