On the classification of gradient Ricci solitons

Abstract

We show that the only shrinking gradient solitons with vanishing Weyl tensor and Ricci tensor satisfying a weak integral condition are quotients of the standard ones $S^n$, $S^{n-1}\times ℝ$ and ${ℝ}^n$. This gives a new proof of the Hamilton–Ivey–Perelman classification of $3$–dimensional shrinking gradient solitons. We also show that gradient solitons with constant scalar curvature and suitably decaying Weyl tensor when noncompact are quotients of ${ℍ}^n$, ${ℍ}^{n-1}\times ℝ$, ${ℝ}^n$, $S^{n-1}\times ℝ$ or $S^n$.