Classification of gradient expanding and steady Ricci solitons

Fei Yang, Shouwen Fang and Liangdi Zhang

Vol. 301 (2019), No. 1, 371–384

DOI: 10.2140/pjm.2019.301.371

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 ${ℝ}^n$. 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 ${ℝ}^n$. For a nontrivial complete noncompact radially Ricci flat (i.e., $Ric\left(\nabla f,\nabla f\right)=0$) gradient steady Ricci soliton with $\int \; <!--nolimits-->|\nabla R{|}^2{e}^{\alpha f}<+\infty$ 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 ${ℝ}^k\times N^{n-k}$ with $1\le k\le n-1$, where $N$ is Einstein with vanishing Ricci curvature.

Keywords

gradient expanding Ricci soliton, gradient steady Ricci soliton

Mathematical Subject Classification 2010

Primary: 53C21, 53C25

Milestones

Received: 21 July 2017

Revised: 13 August 2018

Accepted: 29 October 2018

Published: 16 September 2019