Geometric Structure of Ends of Ricci Shrinkers

TL;DR

This paper investigates the geometric structure of Ricci shrinkers' ends by analyzing blow-up sequences without global curvature bounds, showing their limits split a line, thus generalizing previous results.

Contribution

It extends the $ ext{F}$-convergence theory to analyze blow-up limits of Ricci shrinkers without global curvature assumptions, revealing line splitting in limits.

Findings

Limits of blow-up sequences split a line.

In four dimensions, limits are smooth Ricci shrinkers.

Convergence is in the pointed smooth Cheeger-Gromov sense.

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.