On the classification of four-dimensional gradient Ricci solitons

TL;DR

This paper classifies four-dimensional gradient Ricci solitons, showing that under certain divergence conditions they are either Einstein or quotients of standard spaces, and also provides results for non-compact expanding and steady solitons.

Contribution

It offers new classification results for four-dimensional gradient Ricci solitons under divergence conditions and curvature assumptions, extending existing understanding.

Findings

Gradient shrinking Ricci solitons are either Einstein or quotients of standard spaces under divergence conditions.

Classification of non-compact expanding Ricci solitons with non-negative Ricci curvature.

Results on steady Ricci solitons under specific curvature conditions.

Abstract

In this paper, we prove some classification results for four-dimensional gradient Ricci solitons. For a four-dimensional gradient shrinking Ricci soliton with $div^4Rm^\pm=0$, we show that it is either Einstein or a finite quotient of $\mathbb{R}^4$, $\mathbb{S}^2\times\mathbb{R}^2$ or $\mathbb{S}^3\times\mathbb{R}$. The same result can be obtained under the condition of $div^4W^\pm=0$. We also present some classification results of four-dimensional complete non-compact gradient expanding Ricci soliton with non-negative Ricci curvature and gradient steady Ricci solitons under certain curvature conditions.