Speaker
Description
The RCD condition provides a notion of Riemannian-like metric measure space satisfying synthetic Ricci curvature lower bounds. In this talk I will present a notion of RCD condition localized on an open subset of the ambient space, with variable curvature lower bound, by requiring infinitesimal Hilbertianity and displacement convexity of suitable entropy functionals on neighbourhoods of every point. The main result concerns the stability of this condition with respect to pointed measured Gromov-Hausdorff convergence. This is achieved by employing a Lagrangian approach, local in nature, to the Mosco convergence of the Cheeger energy. This method is based on the construction of good test plans, instead of classically used nonlocal properties of the heat flow that are a priori unavailable in this setting. Furthermore, I will discuss how it is possible to show local essential nonbranching properties through the use of an approximate local Evolution Variational Inequality. The talk is based on joint works with L. Ambrosio, F. Nobili and F. Vitillaro.