On the classification of gradient Ricci solitons

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 Sⁿ, S^n-1 R and Rⁿ . 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 Hⁿ, H^{n 1} R, Rⁿ, S^n-1 R or Sⁿ .