Parole chiave

• Ricci curvature
• Ricci limit
• RCD space

Abstract Starting from the second half of the nineteenth century, it was understood that curvature, which is infinitesimal (geo)metric information on a space, integrates to give global structure results (in spite of its nonlinear nature), specifically yielding topological rigidity of the space. It was further observed by Gromov that a lower bound on the Ricci curvature is the essential ingredient in order to control the number of degrees of freedom at a metric level as well, allowing to compactify (in a very weak sense) the set of $n$-dimensional Riemannian manifolds obeying a Ricci lower bound. It was later understood that singular spaces belonging to this compactification (Ricci limits), are special cases of a more general analytic notion (RCD spaces), more stable with respect to certain natural operations, and thus they also inherit a rich analytic structure, allowing to do calculus on them.

In this talk, based on joint work with Elia Bruè and Daniele Semola, we will review some previously known structural results for Ricci limits and RCDs in the non-collapsed case, as well as some instructive examples and counterexamples, and we will see a new, more elementary proof that Ricci limits of dimension three are generalized manifolds, enjoying in particular uniform contractibility. Our tools, together with some results in geometric topology, give an alternative proof that they are in fact topological manifolds. We will also see a new result for tangent cones in higher dimension. If time allows, we will also mention a new structural theorem for RCDs in dimension three.