Ricci flow with bounded curvature integrals

TL;DR

This paper investigates Ricci flows with bounded integral curvature energies on closed manifolds, proving convergence to smooth orbifolds in four dimensions and extending flows in higher dimensions under certain conditions.

Contribution

It establishes convergence and extension results for Ricci flows with integral curvature bounds, including orbifold singularities and flow extension in higher dimensions.

Findings

In four dimensions, Ricci flow converges to a smooth orbifold with finitely many singularities.

In higher dimensions, similar convergence results hold under additional integral curvature bounds.

Such flows can be extended over finite time intervals as orbifold Ricci flows.

Abstract

In this paper, we study the Ricci flow on a closed manifold and finite time interval $[0,T)~(T < \infty)$ on which certain integral curvature energies are finite. We prove that in dimension four, such flow converges to a smooth Riemannian manifold except for finitely many orbifold singularities. We also show that in higher dimensions, the same assertions hold for a closed Ricci flow satisfying another conditions of integral curvature bounds. Moreover, we show that such flows can be extended over $T$ by an orbifold Ricci flow.