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 be a metric measure space. Define as the inf of lengths of curves in the space of probability measures on , equipped with Wasserstein’s metric, connecting to , wher is the heat kernel at time .
Theorem 1 On Riemannian manifolds, arises from some Riemannian metric . This metric satisfies
So our evolution is tangent to the Ricci flow initially. But it is not a semi-group.
Theorem 2 On -spaces, this evolution is well defined and continuous in time and space.
2. Proof of Theorem 1
Explicit computation. Let denote the heat kernel. For every tangent vector , there is a unique 2-variable function (with vanishing integral) such that (differentiation w.r.t. variable). Then, at ,
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 , Ricci curvature pops out of Bochner’s formula.
3. What is ?
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 , thanks to
Theorem 3 (Sturm, Von Renesse 2005) On Riemannian manifolds, heat flow contracts distances by a factor of .
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 imply ?
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.