On the rigidity of Ricci shrinkers

Abstract

We establish the rigidity of the generalized cylinder $N^n\times {\mathbb{R}}^{m-n}$, or a quotient thereof, in the space of Ricci shrinkers equipped with the pointed Gromov–Hausdorff topology. Here, $N$ is a stable Einstein manifold that has an obstruction of order $3$. The proof is based on a quantitative characterization of the rigidity of compact Ricci shrinkers, a rigidity inequality of mixed orders on generalized cylinders, and the method of contraction and extension. As an application, we prove the uniqueness of the tangent flow for general compact Ricci flows under the assumption that one tangent flow is a generalized cylinder.