AI 中文总结
该研究证明局部黎曼-里奇曲率下界沿格罗莫夫-豪斯多夫收敛的稳定性，通过拉格朗日方法推导弱梯度平行四边形恒等式的稳定性，得到欧氏弱切向量几乎处处存在等应用。
AI 中文摘要
我们证明了局部黎曼-里奇曲率下界沿格罗莫夫-豪斯多夫收敛的稳定性。分析的核心部分是证明弱梯度的平行四边形恒等式的稳定性，该恒等式通过在局部情形下实施[arXiv:2511.13320]中发展的拉格朗日方法得到。作为应用，我们推导出欧氏弱切向量的几乎处处存在性。一个具有独立意义的重要要素是，在足够小的球上沿热流的有效局部演化变分不等式，其余项项取决于热流的衰减速率。这一结果可应用于熵泛函沿局部瓦瑟斯坦插值的强位移凸性，以及局部本质非分支性质。
英文摘要
We establish the stability of local Riemannian Ricci curvature lower bounds along Gromov-Hausdorff convergence. A central part of our analysis is devoted to showing the stability of the parallelogram identity for weak gradients, obtained implementing the Lagrangian approach developed in [arXiv:2511.13320] in the local setting. As an application, we deduce the almost everywhere existence of Euclidean weak tangents. An important ingredient, of independent interest, is an effective local Evolution Variational Inequality along the heat flow on sufficiently small balls, with a remainder term depending on the rate of decay of the flow. This has applications to strong displacement convexity of the Entropy functional along local Wasserstein interpolations and to local essential nonbranching properties.
