L - optimal transportation for ricci flow ∗

ar X iv : 0 80 7 . 41 81 v 1 [ m at h . D G ] 2 5 Ju l 2 00 8 the canonical shrinking soliton associated to a ricci flow

To every Ricci flow on a manifold M over a time interval I ⊂ R, we associate a shrinking Ricci soliton on the space-time M×I . We relate properties of the original Ricci flow to properties of the new higher-dimensional Ricci flow equipped with its own time-parameter. This geometric construction was discovered by consideration of the theory of optimal transportation, and in particular the result...

Optimal Transportation and Monotonic Quantities on Evolving Manifolds

In this note we will adapt Topping’s L-optimal transportation theory for Ricci flow to a more general situation, i.e. to a closed manifold (M, gij(t)) evolving by ∂tgij = −2Sij , where Sij is a symmetric tensor field of (2,0)-type on M . We extend some recent results of Topping, Lott and Brendle, generalize the monotonicity of List’s (and hence also of Perelman’s) W-entropy, and recover the mon...

Ricci Flow: The Foundations via Optimal Transportation

A user’s guide to optimal transport

This text is an expanded version of the lectures given by the first author in the 2009 CIME summer school of Cetraro. It provides a quick and reasonably account of the classical theory of optimal mass transportation and of its more recent developments, including the metric theory of gradient flows, geometric and functional inequalities related to optimal transportation, the first and second ord...

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...