Smoothing metrics on closed Riemannian manifolds through the Ricci flow

Abstract

Under the assumption of the uniform local Sobolev inequality, it is proved that Riemannian metrics with an absolute Ricci curvature bound and a small Riemannian curvature integral bound can be smoothed to having a sectional curvature bound. This partly extends previous a priori estimates of Li (

Small curvature concentration and Ricci flow smoothing

“…Related regularization results for the Ricci flow under critical L n/2 bounds of Rm have previously been studied in [57,37] assuming also pointwise two-sided bounds on |Ric|, and in [52] assuming alternatively a supercritical Ric p , p > n/2 bound. In this note we will also study the flow under local critical L n/2 bounds of Rm, but will instead do so in combination with a Ricci lower bound and control of the local entropy, a localization of Perelman's entropy introduced by Wang [53].…”