Four-Dimensional Gradient Shrinking Solitons with Positive Isotropic Curvature

Abstract

We show that a four-dimensional complete gradient shrinking Ricci soliton with positive isotropic curvature is either a quotient of |$\mathbb{S}^4$| or a quotient of |$\mathbb{S}^3 \times \mathbb R $|⁠. This gives a clean classification result removing the earlier additional assumptions in [14] by Wallach and the second author. The proof also gives a classification result on gradient shrinking Ricci solitons with non-negative isotropic curvature.