Stability of local Riemannian Ricci curvature lower bounds

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.