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RICCI FLOW, ENTROPY AND OPTIMAL TRANSPORTATION
"... Abstract. Let a smooth family of Riemannian metrics g(τ) satisfy the backwards Ricci flow equation on a compact oriented ndimensional manifold M. Suppose two families of normalized nforms ω(τ) ≥ 0 and ˜ω(τ) ≥ 0 satisfy the forwards (in τ) heat equation on M generated by the connection Laplacian ..."
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Cited by 37 (1 self)
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Abstract. Let a smooth family of Riemannian metrics g(τ) satisfy the backwards Ricci flow equation on a compact oriented ndimensional manifold M. Suppose two families of normalized nforms ω(τ) ≥ 0 and ˜ω(τ) ≥ 0 satisfy the forwards (in τ) heat equation on M generated by the connection Laplacian ∆g(τ). If these nforms represent two evolving distributions of particles over M, the minimum rootmeansquare distance W2(ω(τ), ˜ω(τ), τ) to transport the particles of ω(τ) onto those of ˜ω(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 characterize supersolutions to the Ricci flow.
The Canonical Expanding Soliton and Harnack inequalities for Ricci flow
, 2009
"... We introduce the notion of Canonical Expanding Ricci Soliton, and use it to derive new Harnack inequalities for Ricci flow. This viewpoint also gives geometric insight into the existing Harnack inequalities of Hamilton and Brendle. 1 ..."
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Cited by 11 (2 self)
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We introduce the notion of Canonical Expanding Ricci Soliton, and use it to derive new Harnack inequalities for Ricci flow. This viewpoint also gives geometric insight into the existing Harnack inequalities of Hamilton and Brendle. 1