Metric measure spaces satisfying curvature-dimension bounds: geometric and analytical properties

Motivated by a description of lower bounds on Ricci curvature not relying on any smoothness assumption (i.e. synthetic), the theory of metric spaces $(X,mathsf d)$ endowed with a reference measure $mathfrak m$ (i.e. emph{metric measure space}, or m.m.s. for short) has aroused great interest in the last twenty years. The latter theory has grown more and more, addressing several issues:the study of functional and geometric inequalities in structures which are very far from being Euclidean (therefore requiring new non-Riemannian tools), the description of the “closure” of classes of Riemannian manifolds under suitable geometric constraints, the stability of analytic and geometric properties of spaces. The celebrated Lott-Sturm-Villani theory of metric measure spaces cite{Lott-Villani09}, cite{Sturm06I}, cite{Sturm06II} furnishes synthetic notions of a Ricci curvature lower bound $ K$ joint with an upper bound $ N$ on the dimension. Their condition, called the Curvature-Dimension condition and denoted by $mathsf{CD}(K,N)$, is formulated in terms of a modified displacement convexity of an entropy functional along $W_{2}$-Wasserstein geodesics. A weaker variant of $mathsf{CD}(K,N)$, namely the Measure Contraction Property $mathsf{MCP}(K,N)$, was independently introduced by Ohta in cite{Ohta1} and Sturm in cite{Sturm06II}. This thesis is devoted to the study of geometric and analytical properties of metric measure spaces satisfying the above mentioned curvature-dimension bounds; this will done by means of tools from Optimal Transport Theory. The first problem we will consider concerns the establishment of an isoperimetric inequality in m.m.s's satisfying the $mathsf{MCP}(K,N)$ condition. More precisely, in cite{CS19} we prove that if $(X,mathsf d,mathsf{m})$ is an essentially non-branching metric measure space with $mathsf m(X)=1$ and satisfying $mathsf{MCP}(K,N)$, then a sharp isoperimetric inequality `a la L'evy-Gromov holds true. In particular, we identify a family of one-dimensional $mathsf{MCP}(K,N)$-densities, each for every choice of $K,N$, volume $v$ and diameter $D$, not verifying $CD(K,N)$, and having optimal isoperimetric profile for the volume $v$. Measure theoretic rigidity is also obtained. In cite{CCMcCAS} we show that the choice of the squared-distance function as transport cost does not influence the definition of $mathsf{CD}(K,N)$. By denoting with $mathsf{CD}_{p}(K,N)$ the analogous condition but with the cost given by the $p^{th}$ power of the distance, we show that $mathsf{CD}_{p}(K,N)$ are all equivalent conditions for any $p>1$ --- if the space $(X,mathsf d,mathsf m)$ is either non-branching or at least satisfies appropriate versions of the essentially non-branching condition; the needle decomposition or localization technique associated to the L^{1}$-optimal transport problem will be the trait d'union between all the seemingly unrelated $mathsf{CD}_{p}(K,N)$ conditions. The use of $L^1$-Optimal transport is encoded in the $mathsf{CD}^{1}(K,N)$ condition, introduced for the first time in cite{CMi16} to solve the long standing problem of the so-called local-to-global property of $mathsf{CD}(K,N)$. Following the same approach of cite{CMi16}, we will also establish the local-to-global property of $mathsf{CD}_{p}(K,N)$ spaces for $p>1$. Finally, in cite{CGS20} we show that the $mathsf{CD}^{1}(K,N)$ condition can be formulated in terms of displacement convexity along $W_{1}$-geodesics; hence, the two approaches produce the same curvature-dimension condition.Combining this result with the one described previously, we establish that for any $p geq 1$, all the $mathsf{CD}_{p}(K,N)$ conditions, when expressed in terms of displacement convexity, are equivalent, provided the space $X$ satisfies the appropriate essentially non-branching condition.