$L^1$ curvature bounds for Type I Ricci flows

TL;DR

This paper establishes $L^1$ bounds on the Riemann curvature tensor for smooth closed Ricci flows by introducing a maximal symmetry neck concept and a decomposition technique with uniform curvature bounds.

Contribution

It introduces the notion of a neck of maximal symmetry and a decomposition method to obtain $L^1$ curvature bounds in Ricci flows, advancing understanding of curvature behavior.

Findings

$L^1$ bounds of the Riemann curvature tensor are established.

A new concept of a neck of maximal symmetry is introduced.

A decomposition into balls with uniform curvature bounds is proved.

Abstract

We show $L^1$-bounds of the Riemann curvature tensor on a smooth closed $n$-dimensional Ricci flow. To achieve this we introduce the notion of a neck of maximal symmetry, similar to the one in Cheeger-Jiang-Naber and Jiang-Naber and establish a decomposition result by balls with uniform curvature bounds that satisfy an appropriate $(n-2)$-content estimate.