Extending four dimensional Ricci flows with bounded scalar curvature

TL;DR

This paper proves that four-dimensional Ricci flows with bounded scalar curvature on compact manifolds converge to a C^0 orbifold at finite time, and the flow can be extended past this limit using orbifold Ricci flow.

Contribution

It establishes convergence of Ricci flows to orbifolds in four dimensions under bounded scalar curvature and demonstrates flow continuation past singularities via orbifold Ricci flow.

Findings

Flow converges to a C^0 orbifold with finitely many singular points.

Provides estimates on convergence rates near and away from orbifold points.

Shows flow extension past the limit using orbifold Ricci flow.

Abstract

We consider smooth solutions (M,g(t)), 0 <= t <T, to Ricci flow on compact, connected, four dimensional manifolds without boundary. We assume that the scalar curvature is bounded uniformly, and that T is finite. In this case, we show that the metric space (M,d(t)) associated to (M,g(t)) converges uniformly in the C^0 sense to (X,d), as t approaches T, where (X,d) is a C^0 Riemannian orbifold with at most finitely many orbifold points. Estimates on the rate of convergence near and away from the orbifold points are given. We also show that it is possible to continue the flow past (X,d) using the orbifold Ricci flow.