G. Besson THE GEOMETRIZATION CONJECTURE AFTER R ...

402 G. BessonTwo features deserve to be stressed. For the round sphere the flow stops in finite positivetime but has an infinite past. For the hyperbolic manifolds, on the contrary, the flowhas a finite past but an infinite future. We find these aspects in the core of the proofsof the two conjectures. Indeed, for the Poincaré conjecture one is led to (although itis not strictly necessary) show that starting from any Riemannian metric on a simplyconnected3-manifold the flow stops in finite time whereas for Thurston’s conjectureone ought to study the long term behaviour of the evolution.6. The seminal resultOn a compact, connected and simply-connected 3 manifold M the idea is now to startwith a ”shape” and deform it. We hope that, as for the example, the manifold M willcontract to a point and that after rescaling the metric will become more and more round(i.e. constantly curved).This is indeed the case when one adds some assumption on the metric.THEOREM 3 ([10], R. Hamilton, 1982). This scheme works if we start with ametric g 0 which has positive Ricci curvature.This is the seminal theorem of the theory. Clearly it is a step towards the proofof the Poincaré conjecture. The only restriction is important since it is not knownwhether a simply-connected manifold carries a metric of positive Ricci curvature. Theproof is done along the lines mentioned above, the manifold becomes more and moreround while contracting to a point.7. The idea of surgeryThe question is now what happens if we start with a random metric g 0 ? It turns out thatthere are examples showing that the manifold may become singular, i.e. that the scalarcurvature may become infinite on a subset of M. This is the case for the neckpinch:which is a metric on a cylinder which develops a singularity in finite time (see[7]). But it could also be worse,