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 nonRiemannian 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 LottSturmVillani theory of metric measure spaces cite{LottVillani09}, 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 CurvatureDimension 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 curvaturedimension 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 nonbranching metric measure space with $mathsf m(X)=1$ and satisfying $mathsf{MCP}(K,N)$, then a sharp isoperimetric inequality `a la L'evyGromov holds true. In particular, we identify a family of onedimensional $mathsf{MCP}(K,N)$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 squareddistance 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 nonbranching or at least satisfies appropriate versions of the essentially nonbranching 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 socalled localtoglobal property of $mathsf{CD}(K,N)$. Following the same approach of cite{CMi16}, we will also establish the localtoglobal 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 curvaturedimension 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 nonbranching condition.
Metric measure spaces satisfying curvaturedimension bounds: geometric and analytical properties / Santarcangelo, Flavia.  (2020 Sep 24).
Metric measure spaces satisfying curvaturedimension bounds: geometric and analytical properties
Santarcangelo, Flavia
20200924
Abstract
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 nonRiemannian 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 LottSturmVillani theory of metric measure spaces cite{LottVillani09}, 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 CurvatureDimension 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 curvaturedimension 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 nonbranching metric measure space with $mathsf m(X)=1$ and satisfying $mathsf{MCP}(K,N)$, then a sharp isoperimetric inequality `a la L'evyGromov holds true. In particular, we identify a family of onedimensional $mathsf{MCP}(K,N)$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 squareddistance 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 nonbranching or at least satisfies appropriate versions of the essentially nonbranching 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 socalled localtoglobal property of $mathsf{CD}(K,N)$. Following the same approach of cite{CMi16}, we will also establish the localtoglobal 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 curvaturedimension 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 nonbranching condition.File  Dimensione  Formato  

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