On three-dimensional Type I $\kappa$-solutions to the Ricci flow

Abstract

In this short note, we prove that the only simply connected noncompact three-dimensional Type I $\kappa$-solution to the Ricci flow is the shrinking cylinder. This work can be regarded as a generalization of Cao and Chow, and a complement of Ding and Ni. Up to this point, three-dimensional $\kappa$-solutions of Type I are completely classified, and it remains interesting to work further towards Perelman's assertion, that the only remaining possibility of three-dimensional noncompact $\kappa$-solution is the Bryant soliton. Brendle is working to that end. The classification of three-dimensional $\kappa$-solution is of importance to the study of four-dimensional Ricci flows, because of a possible dimension-reduction procedure.