Global regularity of three-dimensional Ricci limit spaces

Abstract

Miles Simon and the second author, in their recent work [Geom. Topol. 25 (2021), pp. 913–948], 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 Simon and the second author in [J. Differential Geometry, to appear] and [Geom. Topol. 25, (2021), pp. 913–948] to improve this local correspondence to a global-local correspondence. That is, we construct a smooth three-manifold M , M , and prove that the entire (weakly) noncollapsed three-dimensional Ricci limit space is homeomorphic to M 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 d g d_g on M , M, where g g is any smooth complete metric on M 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 in the paper by Raphaël Hochard [Short-time existence of the Ricci flow on complete, non-collapsed 3-manifolds with Ricci curvature bounded from below, https://arxiv.org/abs/1603.08726, 2016]. Suppose ( M , g 0 , x 0 ) \left ( M , g_0 , x_0 \right ) is a complete smooth pointed Riemannian three-manifold that is (weakly) noncollapsed and satisfies a lower Ricci bound. Then, given any k ∈ N , k \in \mathbb {N}, we construct a smooth Ricci flow g ( t ) g(t) living on a subset of spacetime that contains, for each j ∈ { 1 , … , k } j \in \left \{1 , \ldots , k \right \} , a cylinder B g 0 ( x 0 , j ) × [ 0 , T j ] \mathbb {B}_{g_0} \left ( x_0 , j \right )\times [0,T_j] , where T j T_j is dependent only on the Ricci lower bound, the (weakly) noncollapsed volume lower bound and the radius j j (in particular independent of k k ) and with the property that g ( 0 ) = g 0 g(0)=g_0 throughout B g 0 ( x 0 , k ) \mathbb {B}_{g_0} \left ( x_0 , k \right ) .