Global Regularity of Three-Dimensional Ricci Limit Spaces

In their recent work [ST17], Miles Simon and the second author established a local bi-Hölder correspondence between weakly noncollapsed Ricci limit spaces in three dimensions and smooth manifolds. In particular, any open ball of finite radius in such a limit space must be bi-Hölder homeomorphic to some open subset of a complete smooth Riemannian three-manifold. In this work we build on the technology from [ST16, ST17] to improve this local correspondence to a global-local correspondence. That is, we construct a smooth three-manifold M, and prove that the entire (weakly) noncollapsed three-dimensional Ricci limit space is homeomorphic to M via a globally-defined homeomorphism that is bi-Hölder once restricted to any compact subset. Here the bi-Hölder regularity is with respect to the distance dg on M, where g is any smooth complete metric on M. A key step in our proof is the construction of local pyramid Ricci flows, existing on uniform regions of spacetime, that are inspired by Hochard’s partial Ricci flows [Hoc16]. Suppose (M, g0, x0) is a complete smooth pointed Riemannian three-manifold that is (weakly) noncollapsed and satisfies a lower Ricci bound. Then, given any k ∈ N, we construct a smooth Ricci flow g(t) living on a subset of spacetime that contains, for each j ∈ {1, . . . , k}, a cylinder Bg0 (x0, j) × [0, Tj ], where Tj is dependent only on the Ricci lower bound, the (weakly) noncollapsed volume lower bound and the radius j (in particular independent of k) and with the property that g(0) = g0 throughout Bg0 (x0, k).