Vol. 301, No. 1, 2019

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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(f,f) = 0) gradient expanding Ricci soliton with nonnegative Ricci curvature is a finite quotient of n. Moreover, we prove that a complete noncompact gradient expanding Ricci soliton with Ric 0 and div4 Rm = 0 is a finite quotient of n. For a nontrivial complete noncompact radially Ricci flat (i.e., Ric(f,f) = 0) gradient steady Ricci soliton with |R|2eαf < + for some α , we show that it is Einstein with vanishing Ricci curvature or a quotient of n or of the product k × Nnk with 1 k 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
Authors
Fei Yang
School of Mathematics and Physics
China University of Geosciences
Wuhan
China
Shouwen Fang
School of Mathematical Science
Yangzhou University
Yangzhou
China
Liangdi Zhang
Center of Mathematical Sciences
Zhejiang University
Hangzhou
China