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 S-n, Sn-1 x 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, Hn-1 x R, R-n, Sn-1 x R or S-n