ℒ-optimal transportation for Ricci flow

L - optimal transportation for ricci flow ∗

We introduce the notion of L-optimal transportation, and use it to construct a natural monotonic quantity for Ricci flow which includes a selection of other monotonicity results, including some key discoveries of Perelman [13] (both related to entropy and to L-length) and a recent result of McCann and the author [11].

Ricci Flow , Entropy and Optimal Transportation ∗

Let a smooth family of Riemannian metrics g(τ) satisfy the backwards Ricci flow equation on a compact oriented n-dimensional manifold M . Suppose two families of normalized n-forms ω(τ) ≥ 0 and ω̃(τ) ≥ 0 satisfy the forwards (in τ) heat equation on M generated by the connection Laplacian ∆g(τ). If these n-forms represent two evolving distributions of particles over M , the minimum root-mean-squa...

Ricci Flow: The Foundations via Optimal Transportation

Optimal Transportation and Ricci Curvature for Metric Measure Spaces

Moreover, we introduce a curvature-dimension condition CD(K, N) being more restrictive than the curvature bound Curv(M,d, m) ≥ K. For Riemannian manifolds, CD(K, N) is equivalent to RicM (ξ, ξ) ≥ K · |ξ|2 and dim(M) ≤ N . Condition CD(K,N) implies sharp version of the Brunn-Minkowski inequality, of the Bishop-Gromov volume comparison theorem and of the Bonnet-Myers theorem. Moreover, it allows ...

Ricci Curvature on Polyhedral Surfaces via Optimal Transportation

The problem of correctly defining geometric objects, such as the curvature, is a hard one in discrete geometry. In 2009, Ollivier defined a notion of curvature applicable to a wide category of measured metric spaces, in particular to graphs. He named it coarse Ricci curvature because it coincides, up to some given factor, with the classical Ricci curvature, when the space is a smooth manifold. ...