On Gradient Shrinking Ricci Solitons with Radial Conditions

Abstract

In this paper, we prove an n-dimensional radially flat gradient shrinking Ricci solitons with $$div^2W(\nabla f,\nabla f)=0$$ is rigid. Moreover, we show that a four-dimensional radially flat gradient shrinking Ricci soliton with $$\text {div}^2W^\pm (\nabla f,\nabla f)=0$$ is either Einstein or a finite quotient of $${\mathbb {R}}^4$$ , $${\mathbb {S}}^2\times {\mathbb {R}}^2$$ or $${\mathbb {S}}^3\times {\mathbb {R}}$$ .