A note on Ricci flow and optimal transportation

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

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].

A Note on Kähler-ricci Flow

Let g(t) with t ∈ [0, T ) be a complete solution to the KaehlerRicci flow: d dt gij̄ = −Rij̄ where T may be ∞. In this article, we show that the curvatures of g(t) is uniformly bounded if the solution g(t) is uniformly equivalet. This result is stronger than the main result in Šešum [6] within the category of Kähler-Ricci flow.

A Note on the Quantum–mechanical Ricci Flow

We obtain Schroedinger quantum mechanics from Perelman’s functional and from the Ricci flow equations of a conformally flat Riemannian metric on a closed 2– dimensional configuration space. We explore links with the recently discussed emergent quantum mechanics.