Ricci flow, entropy and optimal transportation

Source abstract

Let a smooth family of Riemannian metrics $g(\tau)$ satisfy the backwards Ricci flow equation on a compact oriented $n$-dimensional manifold $M$. Suppose two families of normalized $n$-forms $\omega(\tau) \ge 0$ and $\tilde \omega(\tau) \ge 0$ satisfy the forwards (in $\tau$) heat equation on $M$ generated by the connection Laplacian $\Delta_{g(\tau)}$. If these $n$-forms represent two evolving distributions of particles over $M$, the minimum root-mean-square distance $W_2(\omega(\tau),\tilde \omega(\tau),\tau)$ to transport the particles of $\omega(\tau)$ onto those of $\tilde \omega(\tau)$ is shown to be non-increasing as a function of $\tau$, without sign conditions on the curvature of $(M,g(\tau))$. Moreover, this contractivity property is shown to characterize supersolutions to the Ricci flow.