# $\epsilon$-Regularity and Structure of 4-dimensional Shrinking Ricci Solitons

## TL;DR

This paper proves an $ ext{epsilon}$-regularity theorem for 4-dimensional shrinking Ricci solitons, confirming a conjecture and providing insights into their structure and degeneration behavior.

## Contribution

It establishes an $ ext{epsilon}$-regularity result for shrinking Ricci solitons, advancing understanding of their geometric structure and degeneration without entropy bounds.

## Key findings

- Confirmed a conjecture of Cheeger-Tian on $ ext{epsilon}$-regularity.
- Derived structural results on metric degeneration in non-compact solitons.
- Provided detailed analysis of collapsing with bounded curvature in the appendix.

## Abstract

A closed four dimensional manifold cannot possess a non-flat Ricci soliton metric with arbitrarily small $L^2$-norm of the curvature. In this paper, we localize this fact in the case of shrinking Ricci solitons by proving an $\varepsilon$-regularity theorem, thus confirming a conjecture of Cheeger-Tian. As applications, we will also derive structural results concerning the degeneration of the metrics on a family of complete non-compact four dimensional shrinking Ricci solitons without a uniform entropy lower bound. In the appendix, we provide a detailed account of the equivariant good chopping theorem when collapsing with locally bounded curvature happens.