Optimal transport and Perelman’s reduced volume
نویسندگان
چکیده
One of the major tools introduced by Perelman is his reduced volume ̃ V [21, Sect. 7]. This is a certain geometric quantity which is monotonically nondecreasing in time when one has a Ricci flow solution. Perelman’s main use of the reduced volume was to rule out local collapsing in a Ricci flow. Before giving his rigorous proof that ̃ V is monotonic, Perelman gave a heuristic argument [21, Sect. 6]. Given a Ricci flow solution (M, g(τ )) on a compact manifold M , where τ is backward time, Perelman considered the manifold ̃ M = M × SN ×R+ with the Riemannian metric
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