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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).

In the previous lecture, we showed that every $\kappa$-solution generated at least one asymptotic gradient shrinking soliton $t \mapsto (M,g(t))$. This soliton is known to have the following properties:

1. It is ancient: t ranges over $(-\infty,0)$.
2. It is a Ricci flow.
3. M is complete and connected.
4. The Riemann curvature is non-negative (though it could theoretically be unbounded).
5. $\frac{dR}{dt}$ is non-negative.
6. M is $\kappa$-noncollapsed.
7. M is not flat.
8. It obeys the gradient shrinking soliton equation

$\hbox{Ric} + \hbox{Hess}(f) = \frac{1}{2\tau} g$ (1)

for some smooth f.

The main result of this lecture is to classify all such solutions in low dimension:

Theorem 1. (Classification of asymptotic gradient shrinking solitons) Let $t \mapsto (M,g(t))$ be as above, and suppose that the dimension d is at most 3. Then one of the following is true (up to isometry and rescaling):

1. d=2,3 and M is a round shrinking spherical space form (i.e. a round shrinking $S^2$, $S^3$, $\Bbb{RP}^2$, or $S^3/\Gamma$ for some finite group $\Gamma$ acting freely on $S^3$).
2. d=3 and M is the round shrinking cylinder $S^2 \times {\Bbb R}$ or the oriented or unoriented quotient of this cylinder by an involution.

The case d=2 of this theorem is due to Hamilton; the compact d=3 case is due to Ivey; and the full d=3 case was sketched out by Perelman. In higher dimension, partial results towards the full classification (and also relaxing many of the hypotheses 1-8) have been established by Petersen-Wylie, by Ni-Wallach, and by Naber; these papers also give alternate proofs of Perelman’s classification.

To prove this theorem, we induct on dimension. In 1 dimension, all manifolds are flat and so the claim is trivial. We will thus take d=2 or d=3, and assume that the result has already been established for dimension d-1. We will then split into several cases:

1. Case 1: Ricci curvature has a zero eigenvector at some point. In this case we can use Hamilton’s splitting theorem to reduce the dimension by one, at which point we can use the induction hypothesis.
2. Case 2: Manifold noncompact, and Ricci curvature is positive and unbounded. In this case we can take a further geometric limit (using some Toponogov theory on the asymptotics of rays in a positively curved manifold) which is a round cylinder (or quotient thereof), and also a gradient steady soliton. One can easily rule out such an object by studying the potential function of that soliton on a closed loop.
3. Case 3: Manifold noncompact, and Ricci curvature is positive and bounded. Here we shall follow the gradient curves of f using some identities arising from the gradient shrinking soliton equation to get a contradiction.
4. Case 4: Manifold compact, and curvature positive. Here we shall use Hamilton’s rounding theorem to show that one is a round shrinking sphere or spherical space form.

We will follow Morgan-Tian‘s treatment of Perelman’s argument; see also the notes of Kleiner-Lott, the paper of Cao-Zhu, and the book of Chow-Lu-Ni for other treatments of this argument.

We now begin using the theory established in the last two lectures to rigorously extract an asymptotic gradient shrinking soliton from the scaling limit of any given $\kappa$-solution. This will require a number of new tools, including the notion of a geometric limit of pointed Ricci flows $t \mapsto (M, g(t), p)$, which can be viewed as the analogue of the Gromov-Hausdorff limit in the category of smooth Riemannian flows. A key result here is Hamilton’s compactness theorem: a sequence of complete pointed non-collapsed Ricci flows with uniform bounds on curvature will have a subsequence which converges geometrically to another Ricci flow. This result, which one can view as an analogue of the Arzelá-Ascoli theorem for Ricci flows, relies on some parabolic regularity estimates for Ricci flow due to Shi.

Next, we use the estimates on reduced length from the Harnack inequality analysis in Lecture 13 to locate some good regions of spacetime of a $\kappa$-solution in which to do the asymptotic analysis. Rescaling these regions and applying Hamilton’s compactness theorem (relying heavily here on the $\kappa$-noncollapsed nature of such solutions) we extract a limit. Formally, the reduced volume is now constant and so Lecture 14 suggests that this limit is a gradient soliton; however, some care is required to make this argument rigorous. In the next section we shall study such solitons, which will then reveal important information about the original $\kappa$-solution.

Our treatment here is primarily based on Morgan-Tian’s book and the notes of Ye. Other treatments can be found in Perelman’s original paper, the notes of Kleiner-Lott, and the paper of Cao-Zhu. See also the foundational papers of Shi and Hamilton, as well as the book of Chow, Lu, and Ni.