Rigidity and ε-regularity theorems of Ricci shrinkers

In this paper, we study the rigidity and ε-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 ε-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.