Mathematical Research Letters

Volume 15 (2008)

Number 5

On a classification of gradient shrinking solitons

Pages: 941 – 955



Lei Ni (University of California at San Diego)

Nolan Wallach (University of California at San Diego)


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.

Published 1 January 2008