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We now begin the study of (smooth) solutions $t \mapsto (M(t),g(t))$ to the Ricci flow equation

$\frac{d}{dt} g_{\alpha \beta} = - 2 \hbox{Ric}_{\alpha \beta}$, (1)

particularly for compact manifolds in three dimensions. Our first basic tool will be the maximum principle for parabolic equations, which we will use to bound (sub-)solutions to nonlinear parabolic PDE by (super-)solutions, and vice versa. Because the various curvatures $\hbox{Riem}_{\alpha \beta \gamma}^\delta$, $\hbox{Ric}_{\alpha \beta}$, R of a manifold undergoing Ricci flow do indeed obey nonlinear parabolic PDE (see equations (31) from Lecture 1), we will be able to obtain some important lower bounds on curvature, and in particular establishes that the curvature is either bounded, or else that the positive components of the curvature dominate the negative components. This latter phenomenon, known as the Hamilton-Ivey pinching phenomenon, is particularly important when studying singularities of Ricci flow, as it means that the geometry of such singularities is almost completely dominated by regions of non-negative (and often quite high) curvature.

In order to motivate the lengthy and detailed analysis of Ricci flow that will occupy the rest of this course, I will spend this lecture giving a high-level overview of Perelman’s Ricci flow-based proof of the Poincaré conjecture, and in particular how that conjecture is reduced to verifying a number of (highly non-trivial) facts about Ricci flow.

At the risk of belaboring the obvious, here is the statement of that conjecture:

Theorem 1. (Poincaré conjecture) Let M be a compact 3-manifold which is simply connected (i.e. it is connected, and every loop is contractible to a point). Then M is homeomorphic to a 3-sphere $S^3$.

[Unless otherwise stated, all manifolds are assumed to be without boundary.]

I will take it for granted that this result is of interest, but you can read the Notices article of Milnor, the Bulletin article of Morgan, or the Clay Mathematical Institute description of the problem (also by Milnor) for background and motivation for this conjecture. Perelman’s methods also extend to establish further generalisations of the Poincaré conjecture, most notably Thurston’s geometrisation conjecture, but I will focus this course just on the Poincaré conjecture. (On the other hand, the geometrisation conjecture will be rather visibly lurking beneath the surface in the discussion of this lecture.)