Lecture XIX: Heat Flow, Optimal Transport and Ricci Curvature

In this final lecture we wish to explore the connections between Heat Flow, Optimal Transport and Ricci curvature on a smooth compact Riemannian manifold. In doing so we will extend and generalise some of the properties we have proved for the Euclidean space. Let us recall that, in particular, we have established the convexity of the logarithmic entropy in \((\mathcal {P}_2(\mathbb {R}^n),W_2)\) in Theorem 15.16 and the fact that the heat flow (defined as the \(L^2\) gradient flow of the Dirichlet energy) is an \( \operatorname {\mathrm {EVI}}\) gradient flow of the relative entropy in \((\mathcal {P}_2(\mathbb {R}^n),W_2)\) .