## Notes of Karl-Theodor Sturm’s IHP 2014 lecture

Metric measure spaces with synthetic Ricci curvature bounds: state of the art

1. Requirements and definitions

• Equivalent to ${Ricci(\xi,\xi)\geq K|\xi|^2}$ in Riemanian case
• Stable under convergence
• Intrinsic, synthetic.

Bibliography

• Sturm, Lott Villani
• Sturm and many coauthors
• Ambrosio-Gigli-Savaré
• Erbar-Kuwada-Sturm : equivalence of Eulerian and Lagrangian approaches.
• Ambrosio-Mondino-Savaré
• Gigli, Ketterer, Mondino-Naber

2. Dimension independant notions

2.1. ${L^2}$-Wasserstein space

Wasserstein distance on the space of probablity measures is the minimal quadratic cost of couplings.

2.2. ${CD(K,\infty)}$

${CD(K,\infty)}$ requires that Entropy is convex along Wasserstein geodesics. Entropy is ${\int\rho\log\rho\,dm}$ for a measure which has density ${\rho}$ with respect to some background measure ${m}$, ${+\infty}$ otherwise.

2.3. Heat flow

Heat equation arises as the gradient flow for the energy ${\frac{1}{2}(lip_x u)^2\,dm(x)}$ on ${L^2}$ functions, or as the gradient flow for the relative entropy ${\int u\log u\,dm}$ on Wasserstein space. Both approaches coincide under ${CD(K,\infty)}$ (Ambrosio-Gigli-Savaré, after many others in special cases).

2.4. The Riemannian Curvature Dimension condition

${RCD(K,\infty)}$ means ${CD(K,\infty)}$ plus linearity of heat flow. This is stable under convergence. We show that ${RCD(K,\infty)}$ spaces are essentially non branching, i.e. optimal transport almost never uses branching geodesics.

When heat flow is linear, ${RCD(K,\infty)\Leftrightarrow}$ heat flow contracts in Wasserstein distance ${\Leftrightarrow}$ Bakry-Emery’s gradient estimate holds (heat semi-group contracts squared gradient) ${\Leftrightarrow}$ Bochner’s inequality (without dependance on dimension) holds: ${\frac{1}{2}\Delta|\nabla u|^2-\langle\nabla,\nabla\Delta u\rangle \geq K|\nabla u|^2}$.

Bochner’s inequality self improves, this implies an ${L^1}$ grandient estimate (equivalently, heat flow contracts ${\infty}$-Wasserstein distance). It follows that given two points, there exist coupled brownian motions starting there whose distance decreases exponentially.

3. Now we introduce dimension

3.1. ${CD(0,N)}$

Define ${N}$-entropy of a measure with density ${\rho}$ with respect to background measure ${m}$ as

$\displaystyle \begin{array}{rcl} S_N(\mu)=-\int \rho^{1-1/N}\,dm. \end{array}$

Define ${CD(0,N)}$ as convexity of ${S_N}$ in Wasserstein space.

3.2. ${CD^*(K,N)}$

Reduced curvature-dimension condition is a modification of convexity of ${S_N}$ where ${t}$ and ${1-t}$ are replaced with expressions made out of the volume element of the sphere with Ricci curvature ${K}$.

3.3. ${CD^e(K,N)}$

This is ${(K,N)}$-convexity of entropy. For smooth functions ${S}$ on ${{\mathbb R}}$, ${(K,N)}$-convexity means ${Hess S-\frac{1}{N}\nabla S\otimes\nabla S\geq 0}$. This turns out to be equivalent to ${CD^*(K,N)}$ for non-branching spaces.

3.4. Analysis on ${RCD^*(K,N)}$ spaces

With Erbar and Kuwada, we show that heat flow contracts on Wasserstein space. We obtained a Bochner inequality

$\displaystyle \frac{1}{2}\Delta|\nabla u|^2-\langle\nabla,\nabla\Delta u\rangle \geq K|\nabla u|^2 -\frac{1}{N}???$

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 ${RCD^*(K,N)}$ spaces

Gigli obtained the splitting theorem for ${RCD^*(0,N)}$ (if there is a two-sided minimizing geodesic, space is a product). Ketterer proved a maximal diameter theorem for ${RCD^*(K,N)}$ (if space achieves the diameter bound, it is a sperical suspension of a space one dimension less), and a cone theorem (the ${(\kappa,N)}$-cone on a ${RCD^*(N-1,N)}$ space is a ${RCD^*(\kappa N,N+1)}$ space).

Example: the Euclidean cone over the the product of two 2-spheres of radius ${\frac{1}{\sqrt{3}}}$ is a ${RCD^*(0,5)}$-space but not an Alexandrov space.

Mondino-Naber show that ${RCD^*(K,N)}$ spaces have a unique tangent cone, isometric to Euclidean space; at a.e. point.

4. Transformations of ${RCD^*(K,N)}$ spaces

4.1. Conformal change

Change the measure ${m}$ to ${e^V m}$ and the length of curves to

$\displaystyle \begin{array}{rcl} \int|\dot{\gamma}|e^{W(\gamma}\,dt. \end{array}$

I show that a ${RCD^*(K,N)}$ is turned into a ${RCD^*(K',N')}$ space for any ${N'>N}$ and some ${K'}$. One can take ${N'=N}$ in the truly conformal case ${V=NW}$.

4.2. Convexification

The motivation is to apply optimal transport techniques to Neumann Laplacians in non-convex domains. We can turn a non-convex domain into a ${\kappa}$-convex one by a conformal change.