Ricci flow with bounded curvature integrals

Abstract

In this paper, we study the Ricci flow on a closed manifold and finite time interval [0, T ) (T < ∞) 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.