Some integral curvature estimates for the Ricci flow in four dimensions

Abstract

We consider solutions (M,g(t)), 0 <= t <T, to Ricci flow on compact, four dimensional manifolds without boundary. We prove integral curvature estimates which are valid for any such solution. In the case that the scalar curvature is bounded and T is finite, we show that these estimates imply that the (spatial) integral of the square of the norm of the Riemannian curvature is bounded by a constant independent of time t for all 0 <= t<T and that the space time integral over M x [0,T) of the fourth power of the norm of the Ricci curvature is bounded.