Convergence of Ricci flows with bounded scalar curvature

TL;DR

This paper proves convergence and compactness of Ricci flows with bounded scalar curvature, showing smooth convergence away from high codimension singularities and confirming a version of the Hamilton-Tian Conjecture.

Contribution

It establishes a compactness theorem for Ricci flows with bounded scalar curvature and verifies a generalized Hamilton-Tian Conjecture, even in the Riemannian case.

Findings

Ricci flows with bounded scalar curvature converge smoothly outside a codimension ≥ 4 set.

Any sequence of such Ricci flows has a subsequence converging to a smooth metric space away from singularities.

L^{p<2}-curvature bounds are established on time-slices of the flows.

Abstract

In this paper we prove convergence and compactness results for Ricci flows with bounded scalar curvature and entropy. More specifically, we show that Ricci flows with bounded scalar curvature converge smoothly away from a singular set of codimension $\geq 4$. We also establish a general form of the Hamilton-Tian Conjecture, which is even true in the Riemannian case. These results are based on a compactness theorem for Ricci flows with bounded scalar curvature, which states that any sequence of such Ricci flows converges, after passing to a subsequence, to a metric space that is smooth away from a set of codimension $\geq 4$. In the course of the proof, we will also establish $L^{p < 2}$-curvature bounds on time-slices of such flows.