Rigidity of positively curved shrinking Ricci solitons in dimension four

TL;DR

This paper classifies four-dimensional shrinking Ricci solitons with a specific positive curvature condition, showing they are isometric to well-known symmetric spaces or quotients, thus revealing their rigidity.

Contribution

It provides a classification result for 4D shrinking Ricci solitons under a curvature condition, identifying all such solitons as standard symmetric spaces or quotients.

Findings

Classified all 4D shrinking Ricci solitons with $Sec \\geq \\frac{1}{24} R$

Proved they are isometric to known symmetric spaces or quotients

Established rigidity under the given curvature condition

Abstract

We classify four-dimensional shrinking Ricci solitons satisfying $Sec \geq \frac{1}{24} R$, where $Sec$ and $R$ denote the sectional and the scalar curvature, respectively. They are isometric to either $\mathbb{R}^{4}$ (and quotients), $\mathbb{S}^{4}$, $\mathbb{RP}^{4}$ or $\mathbb{CP}^{2}$ with their standard metrics.