Rigidity of Gradient Shrinking Ricci Solitons

TL;DR

This paper establishes rigidity results for gradient shrinking Ricci solitons with divergence-free Riemannian or Weyl tensors, classifying their geometric structure in four dimensions and showing they are either Einstein or specific quotients of known solitons.

Contribution

It proves rigidity theorems for gradient shrinking Ricci solitons with divergence-free tensors, extending classification results in four dimensions.

Findings

Gradient shrinking Ricci solitons with divergence-free Riemannian tensor are rigid.

In 4D, such solitons are either Einstein or quotients of known models.

Results also hold under divergence-free Weyl tensor condition.

Abstract

We prove that a gradient shrinking Ricci soliton with fourth order divergence-free Riemannian tensor is rigid. For the $4$-dimensional case, we show that any gradient shrinking Ricci soliton with fourth order divergence-free Riemannian tensor is either Einstein, or a finite quotient of the Gaussian shrinking soliton $\mathbb{R}^4$, $\mathbb{R}^2\times\mathbb{S}^2$ or the round cylinder $\mathbb{R}\times\mathbb{S}^3$. Under the condition of fourth order divergence-free Weyl tensor, we have the same results.