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Having characterised the structure of $\kappa$-solutions, we now use them to describe the structure of high curvature regions of Ricci flow, as promised back in Lecture 12, in particular controlling their geometry and topology to the extent that surgery will be applied, which we will discuss in the next (and final) lecture of this class.

The material here is drawn largely from Morgan-Tian’s book and Perelman’s first and second papers; see also Kleiner-Lott’s notes and Cao-Zhu’s paper for closely related material. Due to lack of time, some details here may be a little sketchy.

Having classified all asymptotic gradient shrinking solitons in three and fewer dimensions in the previous lecture, we now use this classification, combined with extensive use of compactness and contradiction arguments, as well as the comparison geometry of complete Riemannian manifolds of non-negative curvature, to understand the structure of $\kappa$-solutions in these dimensions, with the aim being to state and prove precise versions of Theorem 1 and Corollary 1 from Lecture 12.

The arguments are particularly simple when the asymptotic gradient shrinking soliton is compact; in this case, the rounding theorems of Hamilton show that the $\kappa$-solution is a (time-shifted) round shrinking spherical space form. This already classifies $\kappa$-solutions completely in two dimensions; the only remaining case is the three-dimensional case when the asymptotic gradient soliton is a round shrinking cylinder (or a quotient thereof by an involution). To proceed further, one has to show that the $\kappa$-solution exhibits significant amounts of curvature, and in particular that one does not have bounded normalised curvature at infinity. This curvature (combined with comparison geometry tools such as the Bishop-Gromov inequality) will cause asymptotic volume collapse of the $\kappa$-solution at infinity. These facts lead to the fundamental Perelman compactness theorem for $\kappa$-solutions, which then provides enough geometric control on such solutions that one can establish the structural theorems mentioned earlier.

The treatment here is a (slightly simplified) version of the arguments in Morgan-Tian’s book, which is based in turn on Perelman’s paper and the notes of Kleiner-Lott (see also the paper of Cao-Zhu for a slightly different treatment of this theory).