Some integral curvature estimates for the Ricci flow in four dimensions

TL;DR

This paper establishes integral curvature bounds for solutions to the Ricci flow on four-dimensional compact manifolds, providing key estimates that hold under bounded scalar curvature and finite time conditions.

Contribution

It introduces new integral curvature estimates for Ricci flow in four dimensions, applicable to solutions with bounded scalar curvature and finite maximal existence time.

Findings

Bounded the space-time integral of the Ricci curvature's fourth power.

Established uniform bounds on the integral of the squared Riemannian curvature.

Proved estimates are valid for any solution with bounded scalar curvature and finite T.

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.