Rigidity and gap theorems for Ricci shrinkers

TL;DR

This paper establishes local Ricci curvature and entropy gap theorems for Ricci shrinkers, depending only on dimension, and applies these results to criteria for removing certain singularities in Ricci flow.

Contribution

It introduces local gap theorems for Ricci shrinkers that depend solely on dimension, extending previous global results and enabling new applications in Ricci flow analysis.

Findings

Local Ricci curvature gap depends only on dimension.

Local $ u$-entropy gap depends only on dimension.

Application to removable Type I singularities in Ricci flow.

Abstract

We prove local versions of the Ricci curvature and $\nu$-entropy gap theorems for Ricci shrinkers, which respectively generalize a previous result of Munteanu-Wang and a prior result of the authors with Ma. The key point is that these local gaps depend only on the dimension and not on the global entropy or any other geometric information of the Ricci shrinker. As an application, we provide a local criterion for removable Type~I singularities of the Ricci flow.