A compactness theorem for complete Ricci shrinkers

TL;DR

This paper establishes a compactness theorem for complete Ricci shrinkers under certain integral and entropy bounds, using a Gauss-Bonnet approach, without requiring pointwise curvature or volume bounds.

Contribution

It proves a new precompactness result for Ricci shrinkers with minimal assumptions, extending previous compactness theorems in geometric analysis.

Findings

Precompactness in orbifold Cheeger-Gromov sense for Ricci shrinkers.

No need for pointwise curvature, volume, or diameter bounds.

In dimension four, an Euler characteristic bound can replace the Riemann bound.

Abstract

We prove precompactness in an orbifold Cheeger-Gromov sense of complete gradient Ricci shrinkers with a lower bound on their entropy and a local integral Riemann bound. We do not need any pointwise curvature assumptions, volume or diameter bounds. In dimension four, under a technical assumption, we can replace the local integral Riemann bound by an upper bound for the Euler characteristic. The proof relies on a Gauss-Bonnet with cutoff argument.