Metric measure spaces with synthetic Ricci curvature bounds: state of the art
1. Requirements and definitions
- Equivalent to in Riemanian case
- Stable under convergence
- Intrinsic, synthetic.
- Sturm, Lott Villani
- Sturm and many coauthors
- Erbar-Kuwada-Sturm : equivalence of Eulerian and Lagrangian approaches.
- Gigli, Ketterer, Mondino-Naber
2. Dimension independant notions
2.1. -Wasserstein space
Wasserstein distance on the space of probablity measures is the minimal quadratic cost of couplings.
requires that Entropy is convex along Wasserstein geodesics. Entropy is for a measure which has density with respect to some background measure , otherwise.
2.3. Heat flow
Heat equation arises as the gradient flow for the energy on functions, or as the gradient flow for the relative entropy on Wasserstein space. Both approaches coincide under (Ambrosio-Gigli-Savaré, after many others in special cases).
2.4. The Riemannian Curvature Dimension condition
means plus linearity of heat flow. This is stable under convergence. We show that spaces are essentially non branching, i.e. optimal transport almost never uses branching geodesics.
When heat flow is linear, heat flow contracts in Wasserstein distance Bakry-Emery’s gradient estimate holds (heat semi-group contracts squared gradient) Bochner’s inequality (without dependance on dimension) holds: .
Bochner’s inequality self improves, this implies an grandient estimate (equivalently, heat flow contracts -Wasserstein distance). It follows that given two points, there exist coupled brownian motions starting there whose distance decreases exponentially.
3. Now we introduce dimension
Define -entropy of a measure with density with respect to background measure as
Define as convexity of in Wasserstein space.
Reduced curvature-dimension condition is a modification of convexity of where and are replaced with expressions made out of the volume element of the sphere with Ricci curvature .
This is -convexity of entropy. For smooth functions on , -convexity means . This turns out to be equivalent to for non-branching spaces.
3.4. Analysis on spaces
With Erbar and Kuwada, we show that heat flow contracts on Wasserstein space. We obtained a Bochner inequality
A sharp Sobolev inequality follows. Garofalo and Mondino showed that the whole Li-Yau program (gradient estimate for the heat kernel) follows. Sharp diameter, Bochner-Gromov estimates follow as well. However, we do not get a sharp isoperimetric inequality.
3.5. Geometry of spaces
Gigli obtained the splitting theorem for (if there is a two-sided minimizing geodesic, space is a product). Ketterer proved a maximal diameter theorem for (if space achieves the diameter bound, it is a sperical suspension of a space one dimension less), and a cone theorem (the -cone on a space is a space).
Example: the Euclidean cone over the the product of two 2-spheres of radius is a -space but not an Alexandrov space.
Mondino-Naber show that spaces have a unique tangent cone, isometric to Euclidean space; at a.e. point.
4. Transformations of spaces
4.1. Conformal change
Change the measure to and the length of curves to
I show that a is turned into a space for any and some . One can take in the truly conformal case .
The motivation is to apply optimal transport techniques to Neumann Laplacians in non-convex domains. We can turn a non-convex domain into a -convex one by a conformal change.