Ricci flow and critical integral curvature pinching

​Smoothing properties of the Ricci flow have provided pointwise pinching results and extensions to pinching in the supercritical L^p, p>n/2 integral curvature cases. However, both here and in other geometric settings, differences arise at the critical, scale-invariant p=n/2 integral norm. I will describe some cases in which generalizations to pinching in this critical integral curvature sense can still be obtained using consequences of the monotonicity of Perelman's W-functional, such as in pinching to space forms and the Gromov--Ruh Theorem. This is joint work with Guofang Wei and Rugang Ye.