Rigidity of four-dimensional Gradient shrinking Ricci solitons

2021/05/22 by Cheng, Xu, Zhou, Detang · 3 citations
#53C20 #53C25 #53E20 #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2105.10744

Abstract

Let (M, g, f) be a 4-dimensional complete noncompact gradient shrinking Ricci soliton with the equation Ric+∇²f=λg, where λ is a positive real number. We prove that if M has constant scalar curvature S=2λ, it must be a quotient of \mathbbS²× ℝ². Together with the known results, this implies that a 4-dimensional complete gradient shrinking Ricci soliton has constant scalar curvature if and only if it is rigid, that is, it is either Einstein, or a finite quotient of Gaussian shrinking soliton ℝ⁴, \BbbS²×ℝ² or \BbbS³×ℝ.

Cited by

• Gradient Shrinking Ricci Solitons and Modified Sectional Curvature
• On Ricci Solitons with Isoparametric Potential Functions
• Rigidity of Five-Dimensional shrinking gradient Ricci solitons

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