Pyramid Ricci Flow in Higher Dimensions

Abstract

In this paper, we construct a pyramid Ricci flow starting with a complete Riemannian manifold $(M^n,g_0)$ that is PIC1, or more generally satisfies a lower curvature bound $K_{I{C}_1}\ge -{\alpha }_0$. That is, instead of constructing a flow on $M\times [0,T]$, we construct it on a subset of space-time that is a union of parabolic cylinders $B_{g_0}(x_0,k)\times [0,T_k]$ for each natural number $k$, where $T_k\downarrow 0$, and prove estimates on the curvature and Riemannian distance. More generally, we construct a pyramid Ricci flow starting with any noncollapsed $I{C}_1$-limit space, and use it to establish that such limit spaces are globally homeomorphic to smooth manifolds via homeomorphisms that are locally bi-Hölder.

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