Ricci Flow , Entropy and Optimal Transportation ∗
نویسندگان
چکیده
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-square distance W2(ω(τ), ω̃(τ), τ) to transport the particles of ω(τ) onto those of ω̃(τ) is shown to be nonincreasing as a function of τ , without sign conditions on the curvature of (M, g(τ)). Moreover, this contractivity property is shown to characterise supersolutions to the Ricci flow.
منابع مشابه
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].
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