Rigidity of four-dimensional Gradient shrinking Ricci solitons

TL;DR

This paper proves that 4-dimensional complete noncompact gradient shrinking Ricci solitons with constant scalar curvature are rigid, specifically being quotients of well-known models like spheres or Euclidean spaces, thus classifying their structure.

Contribution

It establishes a rigidity result for 4D gradient shrinking Ricci solitons with constant scalar curvature, characterizing them as quotients of standard geometric models.

Findings

If scalar curvature is constant and equals 2λ, the manifold is a quotient of S^2×R^2.

Complete 4D gradient shrinking Ricci solitons with constant scalar curvature are either Einstein or quotients of Gaussian solitons.

The classification includes quotients of R^4, S^2×R^2, or S^3×R.

Abstract

Let $(M, g, f)$ be a $4$-dimensional complete noncompact gradient shrinking Ricci soliton with the equation $Ric+\nabla^2f=\lambda g$, where $\lambda$ is a positive real number. We prove that if $M$ has constant scalar curvature $S=2\lambda$, it must be a quotient of $\mathbb{S}^2\times \mathbb{R}^2$. 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 $\Bbb{R}^4$, $\Bbb{S}^{2}\times\Bbb{R}^{2}$ or $\Bbb{S}^{3}\times\Bbb{R}$.