## Formalized Name Sturm Stability of Weak Lott-Sturm-Villani Curvature-Dimension under Pointed Measured Gromov-Hausdorff Convergence ## Formalized 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 \begin{align*} \int_Y r(y,y_0)^2\,d\nu(y)<\infty. \end{align*} 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 \begin{align*} \operatorname{Ent}_q(\nu):=\int_Y \sigma(y)\log\sigma(y)\,dq(y) \end{align*} 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 \begin{align*} \mathcal U_{N,q}(\nu):=-\int_Y \sigma(y)^{1-1/N}\,dq(y) \end{align*} 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. 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 \begin{align*} m_j(B_{d_j}(x_j,1))=1. \end{align*} 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 $N<\infty$. 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 \begin{align*} (X_j,d_j,m_j,x_j)\longrightarrow (X,d,m,x) \end{align*} 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 \begin{align*} \iota_{j,R}:B_{d_j}(x_j,R)\to Z_R, \end{align*} \begin{align*} \iota_R:B_d(x,R)\to Z_R \end{align*} such that the embedded pointed closed balls converge in Hausdorff distance, the measures \begin{align*} (\iota_{j,R})_\#(m_j\!\restriction_{B_{d_j}(x_j,R)}) \end{align*} converge weakly on bounded Borel subsets of $Z_R$ whose boundary has zero limiting measure to \begin{align*} (\iota_R)_\#(m\!\restriction_{B_d(x,R)}), \end{align*} 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 $CD(K,N)$. ## Proof [proofplan] We reduce the stated localized pointed measured Gromov-Hausdorff convergence to the normalized convergence framework used in Sturm's stability theorem for Lott-Sturm-Villani curvature-dimension bounds. The only external input is stated explicitly: along normalized pointed measured Gromov-Hausdorff convergence, compactly supported absolutely continuous endpoints can be approximated with the required entropy convergence, optimal dynamical plans are compact, and the Lott-Sturm-Villani entropy inequalities pass to the limit with the correct lower-semicontinuity direction. After verifying the hypotheses of this stability theorem, we apply it to arbitrary compactly supported absolutely continuous endpoint measures in the limit space and obtain the weak $CD(K,N)$ inequality. [/proofplan] [step:State the external stability theorem used in the argument] We use the following standard theorem of Sturm on the stability of Lott-Sturm-Villani curvature-dimension bounds under normalized measured Gromov-Hausdorff convergence. For any complete separable geodesic metric space $(S,s)$, let $\operatorname{Geo}(S)$ denote the set of constant-speed geodesics $\gamma:[0,1]\to S$ with the topology of uniform convergence, and let $e_t:\operatorname{Geo}(S)\to S$ be the evaluation map \begin{align*} e_t(\gamma)=\gamma(t). \end{align*} [claim:Sturm stability theorem for weak $CD(K,N)$] Let $K\in\mathbb R$ and $N\in(1,\infty]$. Let \begin{align*} (Y_j,r_j,q_j,p_j) \end{align*} be pointed complete separable geodesic metric measure spaces with locally finite Borel measures of full support and normalization \begin{align*} q_j(B_{r_j}(p_j,1))=1. \end{align*} Assume that $(Y_j,r_j,q_j,p_j)$ converge to a pointed complete separable geodesic metric measure space $(Y,r,q,p)$ in normalized pointed measured Gromov-Hausdorff convergence, meaning that the convergence is realized on every bounded ball by isometric embeddings into proper metric spaces, Hausdorff convergence of the embedded closed balls, weak convergence of the restricted reference measures on bounded continuity sets, and tightness on bounded balls. If every $(Y_j,r_j,q_j)$ satisfies the weak Lott-Sturm-Villani condition $CD(K,N)$, then $(Y,r,q)$ satisfies the weak Lott-Sturm-Villani condition $CD(K,N)$. More explicitly, for every compactly supported pair $\nu_0,\nu_1\in\mathcal P_2(Y)$ with $\nu_0\ll q$ and $\nu_1\ll q$, the theorem gives an optimal dynamical plan $\Pi\in\mathcal P(\operatorname{Geo}(Y))$ with endpoint marginals $\nu_0$ and $\nu_1$ such that, writing $\nu_t=(e_t)_\#\Pi$, the following holds. If $N=\infty$, then for every $t\in[0,1]$, \begin{align*} \operatorname{Ent}_{q}(\nu_t)\le (1-t)\operatorname{Ent}_{q}(\nu_0)+t\operatorname{Ent}_{q}(\nu_1)-\frac{K}{2}t(1-t)W_2^2(\nu_0,\nu_1). \end{align*} If $N<\infty$ and $\nu_i=\eta_i q$ for $i\in\{0,1\}$, then for every $t\in[0,1]$, \begin{align*} \mathcal U_{N,q}(\nu_t)\le -\int_{\operatorname{Geo}(Y)}\tau_{K,N}^{(1-t)}(r(\gamma(0),\gamma(1)))\eta_0(\gamma(0))^{-1/N}\,d\Pi(\gamma)-\int_{\operatorname{Geo}(Y)}\tau_{K,N}^{(t)}(r(\gamma(0),\gamma(1)))\eta_1(\gamma(1))^{-1/N}\,d\Pi(\gamma). \end{align*} The finite-dimensional assertion is interpreted with the lower-semicontinuous envelope of the distortion coefficients, so values $+\infty$ are allowed and the displayed inequality is an inequality in the extended real line. [/claim] [proof] This is the standard stability theorem proved by Sturm for curvature-dimension conditions in the Lott-Sturm-Villani formulation. In the proof of that theorem, the normalized measured Gromov-Hausdorff convergence supplies three technical ingredients: approximation of compactly supported absolutely continuous endpoint measures with convergence of the relevant endpoint entropy functionals, compactness of optimal dynamical plans in the localized proper ambient path spaces, and lower semicontinuity of the Boltzmann or finite-dimensional distortion entropy inequality under the varying reference measures. We use this theorem as an external standard result, not as a consequence of the present theorem. [/proof] [guided] The proof uses one external theorem, and it is important to isolate exactly what it says. The parameters are \begin{align*} K\in\mathbb R \end{align*} and \begin{align*} N\in(1,\infty]. \end{align*} The approximating spaces in the theorem are pointed metric measure spaces \begin{align*} (Y_j,r_j,q_j,p_j) \end{align*} with locally finite full-support reference measures and the normalization \begin{align*} q_j(B_{r_j}(p_j,1))=1. \end{align*} They converge to \begin{align*} (Y,r,q,p) \end{align*} in the normalized pointed measured Gromov-Hausdorff sense. The hypothesis on the approximating spaces is exactly \begin{align*} CD(K,N). \end{align*} The conclusion is also exactly \begin{align*} CD(K,N) \end{align*} for \begin{align*} (Y,r,q). \end{align*} Thus, for arbitrary compactly supported endpoint measures \begin{align*} \nu_0,\nu_1\in\mathcal P_2(Y) \end{align*} with \begin{align*} \nu_0\ll q \end{align*} and \begin{align*} \nu_1\ll q, \end{align*} the theorem produces an optimal dynamical plan \begin{align*} \Pi\in\mathcal P(\operatorname{Geo}(Y)) \end{align*} with endpoint marginals \begin{align*} \nu_0 \end{align*} and \begin{align*} \nu_1. \end{align*} Writing \begin{align*} \nu_t=(e_t)_\#\Pi \end{align*} for the time marginals, the case \begin{align*} N=\infty \end{align*} gives, for every \begin{align*} t\in[0,1], \end{align*} the Boltzmann entropy inequality \begin{align*} \operatorname{Ent}_{q}(\nu_t)\le (1-t)\operatorname{Ent}_{q}(\nu_0)+t\operatorname{Ent}_{q}(\nu_1)-\frac{K}{2}t(1-t)W_2^2(\nu_0,\nu_1). \end{align*} In the case \begin{align*} N<\infty \end{align*} with endpoint densities \begin{align*} \nu_i=\eta_iq \end{align*} for \begin{align*} i\in\{0,1\}, \end{align*} the same theorem gives, for every \begin{align*} t\in[0,1], \end{align*} the finite-dimensional distortion inequality \begin{align*} \mathcal U_{N,q}(\nu_t)\le -\int_{\operatorname{Geo}(Y)}\tau_{K,N}^{(1-t)}(r(\gamma(0),\gamma(1)))\eta_0(\gamma(0))^{-1/N}\,d\Pi(\gamma)-\int_{\operatorname{Geo}(Y)}\tau_{K,N}^{(t)}(r(\gamma(0),\gamma(1)))\eta_1(\gamma(1))^{-1/N}\,d\Pi(\gamma). \end{align*} The sign issue in the finite-dimensional inequality is part of the theorem: the lower-semicontinuity statement is formulated for the whole Lott-Sturm-Villani distortion functional, including the negative signs and the lower-semicontinuous envelope of \begin{align*} \tau_{K,N}^{(t)}. \end{align*} Values \begin{align*} +\infty \end{align*} are allowed, and the finite-dimensional inequality is interpreted in the extended real line. Thus no separate dominated-convergence argument is being smuggled into the proof. The external theorem is exactly the result that endpoint approximation, optimal-plan compactness, and the delicate finite-dimensional semicontinuity combine to preserve the weak \begin{align*} CD(K,N) \end{align*} condition. [/guided] [/step] [step:Verify that the present convergence satisfies Sturm's normalized hypotheses] For each $R\in(0,\infty)$, the theorem statement supplies a proper metric space $(Z_R,\delta_R)$ and isometric embeddings \begin{align*} \iota_{j,R}:B_{d_j}(x_j,R)\to Z_R \end{align*} and \begin{align*} \iota_R:B_d(x,R)\to Z_R. \end{align*} The same hypothesis states that the embedded pointed closed balls converge in Hausdorff distance, that the pushed-forward restricted measures \begin{align*} (\iota_{j,R})_\#(m_j\!\restriction_{B_{d_j}(x_j,R)}) \end{align*} converge weakly on bounded Borel continuity sets to \begin{align*} (\iota_R)_\#(m\!\restriction_{B_d(x,R)}), \end{align*} and that these pushed-forward measures are tight on bounded balls. These are precisely the local proper-realization, local weak convergence, and tightness requirements in the normalized pointed measured Gromov-Hausdorff convergence appearing in Sturm's theorem. The measures $m_j$ are locally finite Borel measures with full support by hypothesis, and $m$ is also locally finite with full support by hypothesis. The normalization required by Sturm's theorem is exactly \begin{align*} m_j(B_{d_j}(x_j,1))=1, \end{align*} which is assumed for every $j\in\mathbb N$. The metric spaces $(X_j,d_j)$ and $(X,d)$ are complete, separable, and geodesic by hypothesis. Hence the sequence \begin{align*} (X_j,d_j,m_j,x_j) \end{align*} converges to \begin{align*} (X,d,m,x) \end{align*} in the normalized pointed measured Gromov-Hausdorff sense required by Sturm's stability theorem. [/step] [step:Apply Sturm's theorem to arbitrary endpoint measures in the limit space] Let $\mu_0,\mu_1\in\mathcal P_2(X)$ be compactly supported Borel probability measures such that \begin{align*} \mu_0\ll m \end{align*} and \begin{align*} \mu_1\ll m. \end{align*} By the Radon-Nikodym theorem, there are Borel maps \begin{align*} \rho_i:X\to[0,\infty) \end{align*} for $i\in\{0,1\}$ such that \begin{align*} \mu_i=\rho_i m. \end{align*} The compact support assumption implies that there is $R_0\in(0,\infty)$ such that \begin{align*} \operatorname{supp}\mu_0\cup\operatorname{supp}\mu_1\subset B_d(x,R_0). \end{align*} Thus the endpoint measures are exactly of the class to which the weak $CD(K,N)$ conclusion in Sturm's theorem applies. For every $j\in\mathbb N$, the space $(X_j,d_j,m_j)$ satisfies weak $CD(K,N)$ by hypothesis. The previous step verified all convergence, normalization, support, local finiteness, full-support, completeness, separability, and geodesic hypotheses of Sturm's theorem. Therefore Sturm's stability theorem gives an optimal dynamical plan \begin{align*} \Pi\in\mathcal P(\operatorname{Geo}(X)) \end{align*} from $\mu_0$ to $\mu_1$. Here $\operatorname{Geo}(X)$ denotes the set of constant-speed geodesics $\gamma:[0,1]\to X$ with the topology of uniform convergence, and $e_t:\operatorname{Geo}(X)\to X$ denotes the evaluation map \begin{align*} e_t(\gamma)=\gamma(t). \end{align*} Define \begin{align*} \mu_t:=(e_t)_\#\Pi \end{align*} for $t\in[0,1]$. Since $\Pi$ is an optimal dynamical plan, the curve $t\mapsto\mu_t$ is a constant-speed $W_2$-geodesic from $\mu_0$ to $\mu_1$. [/step] [step:Record the limiting entropy inequality] If $N=\infty$, Sturm's theorem gives, for every $t\in[0,1]$, \begin{align*} \operatorname{Ent}_{m}(\mu_t)\le (1-t)\operatorname{Ent}_{m}(\mu_0)+t\operatorname{Ent}_{m}(\mu_1)-\frac{K}{2}t(1-t)W_2^2(\mu_0,\mu_1). \end{align*} This is the Boltzmann entropy convexity inequality in the weak Lott-Sturm-Villani definition of $CD(K,\infty)$. If $N<\infty$, the same theorem gives, for every $t\in[0,1]$, \begin{align*} \mathcal U_{N,m}(\mu_t)\le -\int_{\operatorname{Geo}(X)}\tau_{K,N}^{(1-t)}(d(\gamma(0),\gamma(1)))\rho_0(\gamma(0))^{-1/N}\,d\Pi(\gamma)-\int_{\operatorname{Geo}(X)}\tau_{K,N}^{(t)}(d(\gamma(0),\gamma(1)))\rho_1(\gamma(1))^{-1/N}\,d\Pi(\gamma). \end{align*} The map $\mathcal U_{N,m}:\mathcal P_2(X)\to[-\infty,0]$ is the finite-dimensional Rényi entropy functional defined from the absolutely continuous density in the Lebesgue decomposition with respect to $m$; the singular part contributes no density term. The displayed inequality is exactly the integrated distortion-coefficient inequality in the weak Lott-Sturm-Villani definition of $CD(K,N)$. [/step] [step:Conclude the weak curvature-dimension condition on the limit space] The compactly supported measures $\mu_0,\mu_1\in\mathcal P_2(X)$ with $\mu_0\ll m$ and $\mu_1\ll m$ were arbitrary. For every such pair, we have constructed an optimal dynamical plan $\Pi\in\mathcal P(\operatorname{Geo}(X))$ whose time marginals form a $W_2$-geodesic and satisfy the required Lott-Sturm-Villani entropy convexity inequality: the Boltzmann entropy inequality when $N=\infty$ and the finite-dimensional distortion-coefficient inequality when $N<\infty$. This is precisely the weak curvature-dimension condition $CD(K,N)$ for $(X,d,m)$. Therefore the limit metric measure space $(X,d,m)$ satisfies $CD(K,N)$. [/step]