Global regularity of three-dimensional Ricci limit spaces

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 three dimensions smooth manifolds. In particular, any open ball of finite radius such space must be homeomorphic to some subset complete Riemannian three-manifold. this we build on technology from author [J. Differential Geometry, appear] 25, 913–948] improve global-local correspondence. That is, construct three-manifold M , M , prove that entire (weakly) three-dimensional is M"> encoding="application/x-tex">M via globally-defined homeomorphism once restricted compact subset. Here regularity with respect distance alttext="d Subscript g"> d g encoding="application/x-tex">d_g encoding="application/x-tex">M, where alttext="g"> encoding="application/x-tex">g metric . A key step our proof construction pyramid flows, existing uniform regions spacetime, are inspired by Hochard’s partial flows paper Raphaël Hochard [Short-time existence flow complete, non-collapsed 3-manifolds curvature bounded below, https://arxiv.org/abs/1603.08726, 2016]. Suppose alttext="left-parenthesis upper comma g 0 x right-parenthesis"> ( 0 x ) encoding="application/x-tex">\left ( , g_0 x_0 \right ) pointed satisfies lower bound. Then, given alttext="k element-of double-struck N k ∈ N encoding="application/x-tex">k \in \mathbb {N}, alttext="g left-parenthesis t ( t stretchy="false">) encoding="application/x-tex">g(t) living spacetime contains, for each alttext="j StartSet 1 ellipsis k EndSet"> j { 1 …<!-- … <mml:mo>} encoding="application/x-tex">j \left \{1 \ldots \} , cylinder alttext="double-struck B Baseline j right-parenthesis times left-bracket T right-bracket"> mathvariant="double-struck">B ×<!-- × stretchy="false">[ T stretchy="false">] encoding="application/x-tex">\mathbb {B}_{g_0} )\times [0,T_j] j"> encoding="application/x-tex">T_j dependent only bound, volume bound alttext="j"> encoding="application/x-tex">j (in particular independent alttext="k"> encoding="application/x-tex">k ) property equals 0"> = encoding="application/x-tex">g(0)=g_0 throughout