2017/08/08 by Zhang, Yongjia
#Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.1708.02341
In this short note, we prove that the only simply connected noncompact three-dimensional Type I κ-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 κ-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 κ-solution is the Bryant soliton. Brendle is working to that end. The classification of three-dimensional κ-solution is of importance to the study of four-dimensional Ricci flows, because of a possible dimension-reduction procedure.