Ricci Flow with Ricci Curvature and Volume Bounded Below

TL;DR

This paper demonstrates that under certain curvature and volume conditions, four-dimensional Ricci flows converge to orbifolds at singularities, with detailed analysis of blowups and curvature bounds, extending understanding of singularity formation.

Contribution

It establishes convergence to orbifold limits at singularities under Ricci curvature and volume bounds, and characterizes tangent flows in higher dimensions.

Findings

Flow converges to a $C^{0}$ orbifold at finite-time singularities.

Type-I blowups at orbifold points converge to flat cones without subsequences.

L^{p} curvature bounds are established for $p<2$ on time slices.

Abstract

We show that a simply-connected closed four-dimensional Ricci flow whose Ricci curvature is uniformly bounded below and whose volume does not approach zero must converge to a $C^{0}$ orbifold at any finite-time singularity, so has an extension through the singularity via orbifold Ricci flow. Moreover, a Type-I blowup of the flow based at any orbifold point converges to a flat cone in the Gromov-Hausdorff sense, without passing to a subsequence. In addition, we prove $L^{p}$ bounds for the curvature tensor on time-slices for any $p<2$. In higher dimensions, we show that every singular point of the flow is a Type-II point, and that any tangent flow at a singular point is a static flow corresponding to a Ricci flat cone.