关于Wasserstein重心几何II：黎曼刚性、本质不分叉与Finsler模型
On the Geometry of Wasserstein Barycenter II: Riemannian Rigidity, Essential Non-Branching, and Finsler Models
浏览论文内容
中文总结 AI 辅助
本文证明Wasserstein重心曲率维数条件BCD(K,N)等价于RCD(K,N)，引入带误差的几乎BCD条件，稳定于Gromov-Hausdorff收敛且蕴含本质不分叉，涵盖非黎曼Finsler空间，并解答了Ambrosio的开放问题。
中文摘要 AI 辅助
Wasserstein重心为度量空间上的概率测度提供了一种加权平均的概念。我们证明，对于$K\in \mathbb R$和$1<N<\infty$，重心曲率维数条件${\rm BCD}(K,N)$等价于${\rm RCD}(K,N)$。这通过有限族测度重心处的熵不等式给出了黎曼曲率维数空间的一个新刻画。作为副产品，我们引入了一个几乎${\rm BCD}$条件，该条件允许熵不等式中存在加性误差。它在度量Gromov-Hausdorff收敛下稳定，并且当误差足够小时蕴含本质不分叉。由此得到的类包含非黎曼Finsler空间。在${\rm BCD}$框架内，这回答了Ambrosio在2018年ICM综述中提出的一个开放问题。我们还进一步用熵误差来界定余切范数的平行四边形恒等式失效的程度。
英文摘要
Wasserstein barycenters provide a notion of weighted mean for probability measures on a metric space. We prove that the barycenter curvature-dimension condition ${\rm BCD}(K,N)$ is equivalent to ${\rm RCD}(K,N)$ for $K\in \mathbb R$ and $1<N<\infty$. This gives a new characterization of Riemannian curvature-dimension spaces by entropy inequalities at barycenters of finite families of measures. As a byproduct, we introduce an almost ${\rm BCD}$ condition that allows an additive error in the entropy inequality. It is stable under measured Gromov-Hausdorff convergence and implies essential non-branching when the error is sufficiently small. The resulting class contains non-Riemannian Finsler spaces. Within the ${\rm BCD}$ framework, this answers an open problem posed by Ambrosio in his 2018 ICM survey. We further bound the failure of the parallelogram identity for cotangent norms in terms of the entropy error.
发表机构
- School of Mathematics, Shandong University（山东大学数学学院）
- School of Mathematical Sciences, University of Science and Technology of China（中国科学技术大学数学科学学院）
机构由 AI 辅助整理，请以论文原文为准。
