Rigidity and {\epsilon}-regularity theorems of Ricci shrinkers

TL;DR

This paper investigates the rigidity and regularity properties of Ricci shrinkers, establishing new theorems that describe their asymptotic structure and curvature behavior at infinity.

Contribution

It proves the rigidity of volume ratios and introduces improved epsilon-regularity theorems for Ricci shrinkers, advancing understanding of their asymptotic geometry.

Findings

Rigidity of asymptotic volume ratio and local volume around a base point.

Enhanced epsilon-regularity theorems for local entropy and curvature.

Under integral curvature conditions, Ricci shrinkers are asymptotic to a cone.

Abstract

In this paper, we study the rigidity and {\epsilon}-regularity theorems of Ricci shrinkers. First we prove the rigidity of the asymptotic volume ratio and local volume around a base point of a non-compact Ricci shrinker. Next we obtain some {\epsilon}-regularity theorems of local entropy and curvature, which improve the previous corresponding results essentially and use them to study the structure of Ricci shrinkers at infinity. Especially, if the curvature of a non-compact Ricci shrinker satisfies some natural integral conditions, then it is asymptotic to a cone.