A local gap theorem for Ricci shrinkers

TL;DR

This paper establishes a local gap theorem for Ricci shrinkers, demonstrating that local geometric conditions near the potential function's minimum point imply the shrinker is flat, thus linking local and global geometry.

Contribution

The paper proves a local gap theorem for Ricci shrinkers based on the local $$-functional, extending previous results that used global functionals.

Findings

If the local $$-functional is close to 0, the Ricci shrinker is the flat Gaussian shrinker.

The result connects local geometric properties to the global structure of Ricci shrinkers.

It generalizes earlier global results to a local setting.

Abstract

We prove a local gap theorem for Ricci shrinkers, which states that if the local $\mu$-functional at scale $1$ on a large ball centered at the minimum point of the potential function is close enough to $0$, then the shrinker must be the flat gaussian shrinker. In relation to our result, Yokota [Yo09,Yo12] proved the same result assuming the global $\mu$-functional to be close enough to $0$. Our result shows an aspect of how the local geometry of a shrinker controls the global geometry, which is also discussed in [LW19,LW20,LW21].