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A Prekopa-Leindler-type inequality for Ricci flow

Cordero-Erausquin, McCann, and Schmuckenschläger [1], [2] have generalized this inequality to Riemannian manifolds. The proof employs techniques from optimal transport theory (see e.g. [9]). In this note, we replace the Riemannian distance by Perelman’s L-distance (cf. [6]). The theory of L-optimal transport was developed in recent work of Topping [8] (see also [4], [5]). Among other things, Topping proved an important monotonicity formula for the L-Wasserstein distance on the space of probability measures. Lott [4] established a convexity property for the entropy along L-Wasserstein geodesics. To fix notation, let M be a compact manifold of dimension n, and let g(t), t ∈ [0, T ], be a one-parameter family of metrics on M . We assume that the metrics g(t) evolve by backward Ricci flow, i.e.