Optimal transport and (dynamic) synthetic curvature bounds on metric measure spaces

This thesis establishes three main results: 1. We extend the result of Lisini (Calc Var Partial Differ Equ 28:85–120, 2007) on the superposition principle for absolutely continuous curves in $p$-Wasserstein spaces to the special case of $p=1$. In contrast to the case of $p>1$, it is not always possible to have lifts on absolutely continuous curves. Therefore, one needs to relax the notion of a lift by considering curves of bounded variation, or shortly BV-curves, and replace the metric speed by the total variation measure. We prove that any BV-curve in a 1-Wasserstein space can be represented by a probability measure on the space of BV-curves which encodes the total variation measure of the Wasserstein curve. In particular, when the curve is absolutely continuous, the result gives a lift concentrated on BV-curves which also characterizes the metric speed. The main theorem is then applied for the characterization of geodesics and the study of the continuity equation in a discrete setting. 2. We obtain an equivalence between the rigidity of Wasserstein contraction along heat flows and the rigidity of Bakry--Émery gradient estimates for Lipschitz functions. Applying results of Ambrosio--Brué--Semola and Han, we show that if an RCD space with Ricci lower bound $K\in[0,\infty)$ admits two distinct points $x,y$ such that the $2$-Wasserstein distance between the associated heat kernels satisfies $W_2(p_t(x,\cdot), p_t(y,\cdot)) = e^{-Kt} d(x,y),$ then the space splits off a line. Moreover, for weighted smooth manifolds, we provide a direct proof of the rigidity theorem for all curvature bounds $K \in \mathbb{R}$. In particular, we characterize a class of weighted Euclidean spaces as the only spaces where the Wasserstein contraction is sharp for all pairs of points. 3. We develop different synthetic notions of Ricci flow in the setting of time-dependent metric measure spaces based on ideas from optimal transport. They are formulated in terms of dynamic convexity and local concavity of the entropy along Wasserstein geodesics on the one hand and in terms of global and short-time asymptotic transport cost estimates for the heat flow on the other hand. We show that these properties characterise smooth (weighted) Ricci flows. Further, we investigate the relation between the different notions in the non-smooth setting of time-dependent metric measure spaces.