Global regularity of three-dimensional Ricci limit spaces - McLeod, AD; Topping, PM (American Mathematical Society, 2022)
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.