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.

About metric2011

metric2011 is a program of Centre Emile Borel, an activity of Institut Henri Poincaré, 11 rue Pierre et Marie Curie, 75005 Paris, France. See http://www.math.ens.fr/metric2011/
This entry was posted in Workshop lecture. Bookmark the permalink.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s