Notes of Nicola Gigli’s lecture

Ricci flow

1. Results

To extend Ricci flow to singular spaces, one needs continuity of Ricci flow with respect to the metric, say, is Gromov-Hausdorff topology. But this fails. We introduce a variant of Ricci flow that has the required continuity property.

Let {(X,d,m)} be a metric measure space. Define {d_t(x,y)} as the inf of lengths of curves in the space {P(X)} of probability measures on {X}, equipped with Wasserstein’s {W_2} metric, connecting {i_t(x)} to {i_t(y)}, wher {i_t(x)} is the heat kernel at time {t}.

Theorem 1 On Riemannian manifolds, {d_t} arises from some Riemannian metric {g_t}. This metric satisfies

\displaystyle  \begin{array}{rcl}  {\partial_t g_t }_{|t=0}=-2Ricci(g_0). \end{array}

So our evolution is tangent to the Ricci flow initially. But it is not a semi-group.

Theorem 2 On {RCD(C,N)}-spaces, this evolution is well defined and continuous in time and space.

2. Proof of Theorem 1

Explicit computation. Let {k_t} denote the heat kernel. For every tangent vector {v\in T_x X}, there is a unique 2-variable function {\phi_t} (with vanishing integral) such that {\nabla k_t\cdot v+\nabla\cdot(\nabla\phi_t k_t)=0} (differentiation w.r.t. {y} variable). Then, at {x},

\displaystyle  \begin{array}{rcl}  g_t(v,v)=\int|\nabla \phi_t|^2 k_t\,dy. \end{array}

The reason is that optimal motions in Wasserstein space go orthogonally to divergence free vectorfields, so they follow gradients. Then the Wasserstein speed equals the Dirichlet energy (with respect to the moving measure).

Now differentiate at {t=0}, Ricci curvature pops out of Bochner’s formula.

3. What is {RCD(C,N)} ?

{RCD\subset CD} is a subclass I introduced with Savaré. It contains metrics whose behaviour is closer to Riemannian. It excludes Finsler metrics, for instance.

Riemannian manifolds belong to {RCD(C,N)}, thanks to

Theorem 3 (Sturm, Von Renesse 2005) On Riemannian manifolds, {Ricci\geq k\Leftrightarrow} heat flow contracts {W_2} distances by a factor of {e^{-kt}}.

4. Questions

I expect the evolved space to be smoother that the initial space. Does it have the same topology ? I expect so. No topological singularities.

In the smooth case, does our evolution preserve a lower Ricci curvature bound ? Of course not, but at least, does {Ricci(g_0)\geq k} imply {Ricci(g_t)\geq F(K,t,n)} ?

Are there computable examples ? What about flat 2D cones ? Under Ricci flow (undefined), they should stay flat. Under our evolution, will the apex smooth away ?

This seems to work starting from a subRiemannian space. A notion of Ricci curvature should arise.

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/
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