# Pyramid Ricci Flow in Higher Dimensions

## TL;DR

This paper introduces a novel construction of pyramid Ricci flows on higher-dimensional manifolds with specific curvature bounds, enabling analysis of limit spaces and their topological properties.

## Contribution

It constructs pyramid Ricci flows on manifolds with PIC1 or lower curvature bounds, extending the flow to noncompact limit spaces and establishing their topological structure.

## Key findings

- Constructed pyramid Ricci flows on manifolds with PIC1 curvature.
- Proved estimates on curvature and distances in the flow.
- Showed limit spaces are homeomorphic to smooth manifolds.

## 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 $K_{IC_1}\geq -\alpha_0$. 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 $IC_1$-limit space, and use it to establish that such limit spaces are globally homeomorphic to smooth manifolds via homeomorphisms that are locally bi-H\"older.