Optimal transportation and monotonic quantities on evolving manifolds (Q611869)

Recently, there has been impressive work on the subject of optimal transport on manifolds whose metrics evolve under the Ricci flow. In particular, in 2009, Peter Topping introduced \({\mathcal{L}}\)-optimal transport for the Ricci flow obtaining monotone quantities along \({\mathcal{L}}\)-Wasserstein geodesics among them Perel'man's \(\mathcal{W}\)-entropy. Independently, Brendle and Lott used the above results to prove the monotonicity of Perel'man's reduced volume. Here, the author considers a Riemannian manifold \((M, g_{ij}(t))\) whose metric evolves in time along a symmetric \(2\)-tensor \({\mathcal{S}}=(S_{ij})\): \[ \frac{\partial g_{ij}}{\partial t}=-2 S_{ij}, \] which obviously includes the case of the Ricci flow (see also R. Müller's work in this direction). Huang extends Topping's \(\mathcal{L}\)-optimal transport to the latter flow, extending his results. In particular, Huang proves the monotonicities of the \(\mathcal{W}\)-entropy and the reduced volume, respectively.