Vol. 301, No. 1, 2019

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Fei Yang, Shouwen Fang and Liangdi Zhang

Vol. 301 (2019), No. 1, 371–384
Abstract

In this paper, we prove some classification theorems for gradient expanding and steady Ricci solitons. We show that a complete noncompact radially Ricci flat (i.e., $Ric\left(\nabla f,\nabla f\right)=0$) gradient expanding Ricci soliton with nonnegative Ricci curvature is a finite quotient of ${ℝ}^{n}\phantom{\rule{0.3em}{0ex}}$. Moreover, we prove that a complete noncompact gradient expanding Ricci soliton with $Ric\ge 0$ and ${div}^{4}Rm=0$ is a finite quotient of ${ℝ}^{n}\phantom{\rule{0.3em}{0ex}}$. For a nontrivial complete noncompact radially Ricci flat (i.e., $Ric\left(\nabla f,\nabla f\right)=0$) gradient steady Ricci soliton with $\int |\nabla R{|}^{2}{e}^{\alpha f}<+\infty$ for some $\alpha \in ℝ$, we show that it is Einstein with vanishing Ricci curvature or a quotient of ${ℝ}^{n}$ or of the product ${ℝ}^{k}×{N}^{n-k}$ with $1\le k\le n-1$, where $N$ is Einstein with vanishing Ricci curvature.