Theorems Sturm Stability of Weak Lott-Sturm-Villani Curvature-Dimension under Pointed Measured Gromov-Hausdorff Convergence Attributions

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Statement

Let $K\in\mathbb R$ and let $N\in(1,\infty]$. For every complete separable metric space $(Y,r)$ and every $y_0\in Y$, let $\mathcal P_2(Y)$ denote the set of Borel probability measures $\nu$ on $Y$ such that

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\begin{align*} \int_Y r(y,y_0)^2\,d\nu(y)<\infty. \end{align*}

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This condition is independent of the chosen point $y_0$. Let $W_2$ denote the quadratic Wasserstein distance induced by the metric in question. For a locally finite Borel measure $q$ on $Y$, define $\operatorname{Ent}_q:\mathcal P_2(Y)\to(-\infty,+\infty]$ by

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\begin{align*} \operatorname{Ent}_q(\nu):=\int_Y \sigma(y)\log\sigma(y)\,dq(y) \end{align*}

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when $\nu=\sigma q+\nu^\perp$ is the Lebesgue decomposition and the integral is finite, and set $\operatorname{Ent}_q(\nu)=+\infty$ otherwise. If $N<\infty$, define $\mathcal U_{N,q}:\mathcal P_2(Y)\to[-\infty,0]$ by

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\begin{align*} \mathcal U_{N,q}(\nu):=-\int_Y \sigma(y)^{1-1/N}\,dq(y) \end{align*}

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for the same Lebesgue decomposition; the singular part contributes no density term. Let $\tau_{K,N}^{(t)}:[0,\infty)\to[0,\infty]$ denote the standard Lott-Sturm-Villani distortion coefficient.

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For each $j\in\mathbb N$, let $(X_j,d_j,m_j,x_j)$ be a pointed complete separable geodesic metric measure space such that $m_j$ is a locally finite Borel measure with full support and

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\begin{align*} m_j(B_{d_j}(x_j,1))=1. \end{align*}

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Assume that each $(X_j,d_j,m_j)$ satisfies the weak Lott-Sturm-Villani curvature-dimension condition $CD(K,N)$: for every compactly supported pair $\mu_0,\mu_1\in\mathcal P_2(X_j)$ with $\mu_0\ll m_j$ and $\mu_1\ll m_j$, there is an optimal dynamical plan on constant-speed geodesics joining them such that the corresponding Lott-Sturm-Villani entropy convexity inequality holds, namely the $K$-convex Boltzmann entropy inequality when $N=\infty$ and the integrated distortion-coefficient inequality involving $\mathcal U_{N,m_j}$ and $\tau_{K,N}^{(t)}$ when $N<\infty$.

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Let $(X,d,m,x)$ be a pointed complete separable geodesic metric measure space such that $m$ is a locally finite Borel measure with full support. Assume that

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\begin{align*} (X_j,d_j,m_j,x_j)\longrightarrow (X,d,m,x) \end{align*}

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in pointed measured Gromov-Hausdorff convergence in the following localized measured sense: for every $R\in(0,\infty)$ there are a proper metric space $(Z_R,\delta_R)$ and isometric embeddings

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\begin{align*} \iota_{j,R}:B_{d_j}(x_j,R)\to Z_R, \end{align*}

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\begin{align*} \iota_R:B_d(x,R)\to Z_R \end{align*}

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such that the embedded pointed closed balls converge in Hausdorff distance, the measures

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\begin{align*} (\iota_{j,R})_\#(m_j\!\restriction_{B_{d_j}(x_j,R)}) \end{align*}

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converge weakly on bounded Borel subsets of $Z_R$ whose boundary has zero limiting measure to

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\begin{align*} (\iota_R)_\#(m\!\restriction_{B_d(x,R)}), \end{align*}

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and these pushed-forward measures are tight on bounded balls. Then the limit metric measure space $(X,d,m)$ satisfies the weak Lott-Sturm-Villani curvature-dimension condition $CD(K,N)$.

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