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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).

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.

We now turn to the theory of parabolic Harnack inequalities, which control the variation over space and time of solutions to the scalar heat equation

$u_t = \Delta u$ (1)

which are bounded and non-negative, and (more pertinently to our applications) of the curvature of Ricci flows

$g_t = -2\hbox{Ric}$ (2)

whose Riemann curvature $\hbox{Riem}$ or Ricci curvature $\hbox{Ric}$ is bounded and non-negative. For instance, the classical parabolic Harnack inequality of Moser asserts, among other things, that one has a bound of the form

$u(t_1,x_1) \leq C(t_1,x_1,t_0,x_0,T_-,T_+,M) u(t_0,x_0)$ (3)

whenever $u: [T_-,T_+] \times M \to {\Bbb R}^+$ is a bounded non-negative solution to (1) on a complete static Riemannian manifold M of bounded curvature, $(t_1,x_1), (t_0,x_0) \in [T_-,T_+] \times M$ are spacetime points with $t_1 < t_0$, and $C(t_1,x_1,t_0,x_0,T_-,T_+,M)$ is a constant which is uniformly bounded for fixed $t_1,t_0,T_-,T_+,M$ when $x_1,x_0$ range over a compact set. (The even more classical elliptic Harnack inequality gives (1) in the steady state case, i.e. for bounded non-negative harmonic functions.) In terms of heat kernels, one can view (1) as an assertion that the heat kernel associated to $(t_0,x_0)$ dominates (up to multiplicative constants) the heat kernel at $(t_1,x_1)$.

The classical proofs of the parabolic Harnack inequality do not give particularly sharp bounds on the constant $C(t_1,x_1,t_0,x_0,T_-,T_+,M)$. Such sharp bounds were obtained by Li and Yau, especially in the case of the scalar heat equation (1) in the case of static manifolds of non-negative Ricci curvature, using Bochner-type identities and the scalar maximum principle. In fact, a stronger differential version of (3) was obtained which implied (3) by an integration along spacetime curves (closely analogous to the ${\mathcal L}$-geodesics considered in earlier lectures). These bounds were particularly strong in the case of ancient solutions (in which one can send $T_- \to -\infty$). Subsequently, Hamilton applied his tensor-valued maximum principle together with some remarkably delicate tensor algebra manipulations to obtain Harnack inequalities of Li-Yau type for solutions to the Ricci flow (2) with bounded non-negative Riemannian curvature. In particular, this inequality applies to the $\kappa$-solutions introduced in the previous lecture.

In this current lecture, we shall discuss all of these inequalities (although we will not give the full details for the proof of Hamilton’s Harnack inequality, as the computations are quite involved), and derive several important consequences of that inequality for $\kappa$-solutions. The material here is based on several sources, including Evans’ PDE book, Müller’s book, Morgan-Tian’s book, the paper of Cao-Zhu, and of course the primary source papers mentioned in this article.

In previous lectures, we have established (modulo some technical details) two significant components of the proof of the Poincaré conjecture: finite time extinction of Ricci flow with surgery (Theorem 4 of Lecture 2), and a $\kappa$-noncollapsing of Ricci flows with surgery (which, except for the surgery part, is Theorem 2 of Lecture 7). Now we come to the heart of the entire argument: the topological and geometric control of the high curvature regions of a Ricci flow, which is absolutely essential in order for one to define surgery on these regions in order to move the flow past singularities. This control is intimately tied to the study of a special type of Ricci flow, the $\kappa$-solutions to the Ricci flow equation; we will be able to use compactness arguments (as well as the $\kappa$-noncollapsing results already obtained) to deduce control of high curvature regions of arbitrary Ricci flows from similar control of $\kappa$-solutions. A secondary compactness argument lets us obtain that control of $\kappa$-solutions from control of an even more special type of solution, the gradient shrinking solitons that we already encountered in Lecture 8.

[Even once one has this control of high curvature regions, the proof of the Poincaré conjecture is still not finished; there is significant work required to properly define the surgery procedure, and then one has to show that the surgeries do not accumulate in time, and also do not disrupt the various monotonicity formulae that we are using to deduce finite time extinction, $\kappa$-noncollapsing, etc. But the control of high curvature regions is arguably the largest single task one has to establish in the entire proof.]

The next few lectures will be devoted to the analysis of $\kappa$-solutions, culminating in Perelman’s topological and geometric classification (or near-classification) of such solutions (which in particular leads to the canonical neighbourhood theorem for these solutions, which we will briefly discuss below). In this lecture we shall formally define the notion of a $\kappa$-solution, and indicate informally why control of such solutions should lead to control of high curvature regions of Ricci flows. We’ll also outline the various types of results that we will prove about $\kappa$-solutions.

Our treatment here is based primarily on the book of Morgan and Tian.