Geometric Structure of Ends of Ricci Shrinkers

Abstract

We study blow-up sequences of Ricci shrinkers without global curvature assumptions based at points $q$ at which the scalar curvature satisfies a Type I bound, proving that their $\mathbb{F}$-limits split a line. In the four-dimensional case these limits are smooth Ricci shrinkers and the convergence is in the pointed smooth Cheeger-Gromov sense. As a consequence, limits along the integral curve of $\nabla f$ starting at such a point $q$ split a line. This generalises known results about the geometry of ends of Ricci shrinkers that relied on global curvature bounds. To obtain our results, we extend the $\mathbb{F}$-convergence theory from Bamler and Li-Wang.