A note on the compactness theorem for 4d Ricci shrinkers

TL;DR

This paper refines the compactness theorem for 4-dimensional Ricci shrinkers by removing certain assumptions, leveraging recent estimates to broaden the theorem's applicability in geometric analysis.

Contribution

It demonstrates that the upper bound on the Euler characteristic and the lower bound on the gradient of the potential are unnecessary assumptions for the compactness theorem.

Findings

The assumptions on Euler characteristic and potential gradient can be removed.

The proof utilizes recent Cheeger-Naber estimates.

The result generalizes the compactness theorem for 4d Ricci shrinkers.

Abstract

In arXiv:1005.3255 we proved an orbifold Cheeger-Gromov compactness theorem for complete 4d Ricci shrinkers with a lower bound for the entropy, an upper bound for the Euler characterisic, and a lower bound for the gradient of the potential at large distances. In this note, we show that the last two assumptions in fact can be removed. The key ingredient is a recent estimate of Cheeger-Naber arXiv:1406.6534.