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 andῶ(τ) ≥0 satisfy the forwards (in τ) heat equation on M generated by the connection Laplacian Δg(τ). If these n-forms represent two evolving distributions of particles over M, the minimum root-mean-square distance W 2 (ω(τ),ῶ(τ),τ) to transport the particles of ω(τ)onto those of ῶ(τ) is shown to be non-increasing as a function of τ, without sign conditions on the curvature of (M,g(τ)). Moreover, this contractivity property is shown to characterize supersolutions to the Ricci flow.