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

@article{Ambrosio2015RiemannianRC, title={Riemannian Ricci curvature lower bounds in metric measure spaces with -finite measure}, author={Luigi Ambrosio and Nicola Gigli and Andrea Mondino and Tapio Rajala}, journal={Transactions of the American Mathematical Society}, year={2015}, volume={367}, pages={4661-4701} }

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 class of spaces denoted by RCD(K,∞). This notion relates the CD(K,N) theory of Sturm and Lott-Villani, in the case N = ∞, to the Bakry-Emery approach. In (4) the RCD(K,∞) property is defined in three equivalent ways and several properties of RCD(K,∞) spaces, including the regularization properties… Expand

#### 175 Citations

On quotients of spaces with Ricci curvature bounded below

- Mathematics
- 2017

Abstract Let ( M , g ) be a smooth Riemannian manifold and G a compact Lie group acting on M effectively and by isometries. It is well known that a lower bound of the sectional curvature of ( M , g )… Expand

Metric measure spaces with Riemannian Ricci curvature bounded from below

- Mathematics
- 2014

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… Expand

Structure theory of metric measure spaces with lower Ricci curvature bounds

- Mathematics
- 2014

We prove that a metric measure space (X,d,m) satisfying finite dimensional lower Ricci curvature bounds and whose Sobolev space W1,2 is Hilbert is rectifiable. That is, a RCD∗(K,N)-space is… Expand

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… Expand

Convergence of pointed non-compact metric measure spaces and stability of Ricci curvature bounds and heat flows

- Mathematics
- 2015

Aim of this paper is to discuss convergence of pointed metric measure spaces in absence of any compactness condition. We propose various definitions, show that all of them are equivalent and that for… Expand

Rigidity and almost rigidity of Sobolev inequalities on compact spaces with lower Ricci curvature bounds

- Mathematics
- 2021

We prove that if M is a closed n-dimensional Riemannian manifold, n ≥ 3, with Ric ≥ n − 1 and for which the optimal constant in the critical Sobolev inequality equals the one of the n-dimensional… Expand

Harmonic functions with polynomial growth on manifolds with nonnegative Ricci curvature

- Mathematics
- 2021

Suppose (M, g) is a Riemannian manifold having dimension n, nonnegative Ricci curvature, maximal volume growth and unique tangent cone at infinity. In this case, the tangent cone at infinity C(X) is… Expand

Measure rigidity of Ricci curvature lower bounds

- Mathematics
- 2016

Abstract The measure contraction property, MCP for short, is a weak Ricci curvature lower bound conditions for metric measure spaces. The goal of this paper is to understand which structural… Expand

Topics on calculus in metric measure spaces

- Mathematics
- 2015

This thesis concerns in some topics on calculus in metric measure spaces, in connection with optimal transport theory and curvature-dimension conditions. We study the continuity equations on metric… Expand

A Talenti-type comparison theorem for RCD(K, N) spaces and applications

- 2020

We prove pointwise and L-gradient comparison results for solutions to elliptic Dirichlet problems defined on open subsets of a (possibly non-smooth) space with positive Ricci curvature (more… Expand

#### References

SHOWING 1-10 OF 40 REFERENCES

Metric measure spaces with Riemannian Ricci curvature bounded from below

- Mathematics
- 2014

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… Expand

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… Expand

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… Expand

On the differential structure of metric measure spaces and applications

- Mathematics
- 2012

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… Expand

Ricci curvature of Markov chains on metric spaces

- Mathematics
- 2007

Abstract We define the coarse Ricci curvature of metric spaces in terms of how much small balls are closer (in Wasserstein transportation distance) than their centers are. This definition naturally… Expand

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… Expand

Improved geodesics for the reduced curvature-dimension condition in branching metric spaces

- Mathematics
- 2012

In this note we show that in metric measure spaces satisfying the reduced curvature-dimension condition CD*(K,N) we always have geodesics in the Wasserstein space of probability measures that satisfy… Expand

Eulerian Calculus for the Displacement Convexity in the Wasserstein Distance

- Mathematics, Computer Science
- 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. Expand

Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below

- Mathematics
- 2014

This paper is devoted to a deeper understanding of the heat flow and to the refinement of calculus tools on metric measure spaces $(X,\mathsf {d},\mathfrak {m})$. Our main results are: A general… Expand

Interpolated measures with bounded density in metric spaces satisfying the curvature-dimension conditions of Sturm

- Mathematics
- 2011

We construct geodesics in the Wasserstein space of probability measure along which all the measures have an upper bound on their density that is determined by the densities of the endpoints of the… Expand