On a classification of gradient shrinking solitons

The main purpose of this article is to provide an alternate proof to a result of Perelman on gradient shrinking solitons. Moreover in dimension three our proof generalizes Perelman’s result by removing the $\kappa$-non-collapsing assumption and allowing general curvature growth. The method also allows us to prove a classification result on gradient shrinking solitons with vanishing Weyl curvature tensor in high dimensions, which includes the rotationally symmetric ones.

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Published 1 January 2008