On the rigidity of Wasserstein contraction along heat flows
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909704942256128 |
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| author | Li, Zhenhao |
| author_facet | Li, Zhenhao |
| contents | We establish 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_02280 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the rigidity of Wasserstein contraction along heat flows Li, Zhenhao Metric Geometry Differential Geometry 53C23, 49Q22, 53C24 We establish 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. |
| title | On the rigidity of Wasserstein contraction along heat flows |
| topic | Metric Geometry Differential Geometry 53C23, 49Q22, 53C24 |
| url | https://arxiv.org/abs/2505.02280 |