You are currently browsing the tag archive for the ‘Sacks-Uhlenbeck theory’ tag.

Returning (perhaps anticlimactically) to the subject of the Poincaré conjecture, recall from Lecture 2 that one of the key pillars of the proof of that conjecture is the finite time extinction result (see Theorem 4 from that lecture), which asserted that if a compact Riemannian 3-manifold (M,g) was initially simply connected, then after a finite amount of time evolving via Ricci flow with surgery, the manifold will be empty.

In this lecture and the next few, we will describe some of the key ideas used to prove this theorem. We will not be able to completely establish this theorem at present, because we do not have a full definition of “surgery”, but we will be able to establish some partial results, and indicate (in informal terms) how to cope with the additional technicalities caused by the surgery procedure. Hopefully, if time permits later in the class, once we have studied the surgery process, I will be able to revisit this material and flesh out these technicalities a bit more.

The proof of finite time extinction proceeds in several stages. The first stage, which was already accomplished in the previous lecture (in the absence of surgery, at least), is to establish lower bounds on the least scalar curvature $R_{\min}$. The next stage, which we discuss in this lecture, is to show that the second homotopy group $\pi_2(M)$ of the manifold must become extinct in finite time, thus all immersed copies of the 2-sphere $S^2$ in M(t) for sufficiently large t must be contractible to a point. The third stage is to show that the third homotopy group $\pi_3(M)$ also becomes extinct so that all immersed copies of the 3-sphere $S^3$ in M are similarly contractible. The final stage, which uses homology theory, is to show that a non-empty 3-manifold cannot have $\pi_1(M), \pi_2(M), \pi_3(M)$ simultaneously trivial, thus yielding the desired claim (note that a simply connected manifold has trivial $\pi_1(M)$ by definition; also, from Exercise 2 of Lecture 2 we see that all components of M remain simply connected even after surgery).

More precisely, in this lecture we will discuss (most of) the proof of

Theorem 1. (Finite time extinction of $\pi_2(M)$) Let $t \mapsto (M(t),g(t))$ be a Ricci flow with surgery on compact 3-manifolds with $t \in [0,+\infty)$, with M(0) containing no embedded copy of $\Bbb{RP}^2$ with trivial normal bundle. Then for all sufficiently large t, $\pi_2(M(t))$ is trivial (or more precisely, every connected component of M(t) has trivial $\pi_2$).

The technical assumption about having no copy of $\Bbb{RP}^2$ with trivial normal bundle is needed solely in order to apply the known existence theory for Ricci flow with surgery (see Theorem 2 from Lecture 2).

The intuition for this result is as follows. From the Gauss-Bonnet theorem (and the fact that the Euler characteristic $\chi(S^2)=V-E+F=2$ of the sphere is positive), we know that 2-spheres tend to have positive (Gaussian) curvature on the average, which should make them shrink under Ricci flow. (Here I am conflating Gaussian curvature with Ricci curvature; however, by restricting to a special class of 2-spheres, namely minimal surfaces, one can connect the two notions of curvature to each other (and to scalar curvature) quite nicely.) On the other hand, the presence of negative scalar curvature can counteract this by expanding these spheres. But the lower bounds on scalar curvature tell us that the negativity of scalar curvature becomes weakened over time, and it turns out that the shrinkage caused by the Gauss-Bonnet theorem eventually dominates and sends the area of all minimal immersed 2-spheres into zero, at which point one can conclude the triviality of $\pi_2(M)$ by the Sacks-Uhlenbeck theory of minimal 2-spheres.

The arguments here are drawn from the book of Morgan-Tian and from the paper of Colding-Minicozzi. The idea of using minimal surfaces to force disappearance of various topological structures under Ricci flow originates with Hamilton (who used 2-torii instead of 2-spheres, but the idea is broadly the same).