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

@article{Ambrosio2015BakrymeryCC,
  title={Bakry-{\'E}mery curvature-dimension condition and Riemannian Ricci curvature bounds},
  author={Luigi Ambrosio and Nicola Gigli and Giuseppe Savar{\'e}},
  journal={Annals of Probability},
  year={2015},
  volume={43},
  pages={339-404}
}
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 from a strongly local Dirichlet form E admitting a Carre du champ Γ in a Polish measure space (X,m) and a canonical distance dE that induces the original topology of X. We first characterize the distinguished class of Riemannian Energy measure spaces, where E coincides with the Cheeger energy… Expand
Riemannian Ricci curvature lower bounds in metric measure spaces with -finite measure
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 correspondingExpand
Metric measure spaces with Riemannian Ricci curvature bounded from below
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 FinslerExpand
Diffusion, Optimal Transport and Ricci Curvature for Metric Measure Space
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 andExpand
Bakry-Émery Conditions on Almost Smooth Metric Measure Spaces
Abstract In this short note, we give a sufficient condition for almost smooth compact metric measure spaces to satisfy the Bakry-Émery condition BE(K, N). The sufficient condition is satisfied forExpand
Diffusion, optimal transport and Ricci curvature
Starting from the pioneering paper of Otto-Villani [11], the link between optimal transport and Ricci curvature in smooth Riemannian geometry has been deeply studied [8, 13]. Among the variousExpand
Measure rigidity of Ricci curvature lower bounds
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 structuralExpand
Geometry and analysis of Dirichlet forms (II)
Abstract Given a regular, strongly local Dirichlet form E , under assumption that the lower bound of the Ricci curvature of Bakry–Emery, the local doubling and local Poincare inequalities areExpand
Structure theory of metric measure spaces with lower Ricci curvature bounds
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 isExpand
On the Geometry of Metric Measure Spaces with Variable Curvature Bounds
Motivated by a classical comparison result of J. C. F. Sturm, we introduce a curvature-dimension condition CD(k, N) for general metric measure spaces, variable lower curvature bound $$k$$k and upperExpand
Stratified spaces and synthetic Ricci curvature bounds
We prove that a compact stratied space satises the Riemannian curvature-dimension condition RCD(K, N) if and only if its Ricci tensor is bounded below by K ∈ R on the regular set, the cone angleExpand
...
1
2
3
4
5
...

References

SHOWING 1-10 OF 69 REFERENCES
Metric measure spaces with Riemannian Ricci curvature bounded from below
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 FinslerExpand
Riemannian Ricci curvature lower bounds in metric measure spaces with -finite measure
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 correspondingExpand
Ricci curvature for metric-measure spaces via optimal transport
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 theExpand
Geometry and analysis of Dirichlet forms (II)
Abstract Given a regular, strongly local Dirichlet form E , under assumption that the lower bound of the Ricci curvature of Bakry–Emery, the local doubling and local Poincare inequalities areExpand
Convergence of pointed non-compact metric measure spaces and stability of Ricci curvature bounds and heat flows
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 forExpand
Weak curvature conditions and functional inequalities
We give sufficient conditions for a measured length space (X, d,ν) to admit local and global Poincare inequalities, along with a Sobolev inequality. We first introduce a condition DM on (X, d,ν) ,dExpand
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 SobolevExpand
A Riemannian interpolation inequality à la Borell, Brascamp and Lieb
Abstract.A concavity estimate is derived for interpolations between L1(M) mass densities on a Riemannian manifold. The inequality sheds new light on the theorems of Prékopa, Leindler, Borell,Expand
Eulerian Calculus for the Displacement Convexity in the Wasserstein Distance
TLDR
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
Local Poincaré inequalities from stable curvature conditions on metric spaces
We prove local Poincaré inequalities under various curvature-dimension conditions which are stable under the measured Gromov–Hausdorff convergence. The first class of spaces we consider is that ofExpand
...
1
2
3
4
5
...