LAUSR

In this paper, we construct a pyramid Ricci flow starting with a complete Riemannian manifold $$(M^n,g_0)$$ ( M n , g 0 ) that is PIC1, or more generally satisfies a lower curvature bound $${\mathrm {K_{IC_1}}}\ge -\alpha _0$$ K IC 1 ≥ - α 0 . That is, instead of constructing a flow on $$M\times [0,T]$$ M × [ 0 , T ] , we construct it on a subset of space-time that is a union of parabolic cylinders $${{\mathbb {B}}}_{g_0}(x_0,k)\times [0,T_k]$$ B g 0 ( x 0 , k ) × [ 0 , T k ] for each $$k\in {{\mathbb {N}}}$$ k ∈ N , where $$T_k\downarrow 0$$ T k ↓ 0 , and prove estimates on the curvature and Riemannian distance. More generally, we construct a pyramid Ricci flow starting with any noncollapsed $$\mathrm {IC_1}$$ IC 1 -limit space, and use it to establish that such limit spaces are globally homeomorphic to smooth manifolds via homeomorphisms that are locally bi-Hölder.