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We now set aside our discussion of the finite time extinction results for Ricci flow with surgery (Theorem 4 from Lecture 2), and turn instead to the main portion of Perelman’s argument, which is to establish the global existence result for Ricci flow with surgery (Theorem 2 from Lecture 2), as well as the discreteness of the surgery times (Theorem 3 from Lecture 2).
As mentioned in Lecture 1, local existence of the Ricci flow is a fairly standard application of nonlinear parabolic theory, once one uses de Turck’s trick to transform Ricci flow into an explicitly parabolic equation. The trouble is, of course, that Ricci flow can and does develop singularities (indeed, we have just spent several lectures showing that singularities must inevitably develop when certain topological hypotheses (e.g. simple connectedness) or geometric hypotheses (e.g. positive scalar curvature) occur). In principle, one can use surgery to remove the most singular parts of the manifold at every singularity time and then restart the Ricci flow, but in order to do this one needs some rather precise control on the geometry and topology of these singular regions. (In particular, there are some hypothetical bad singularity scenarios which cannot be easily removed by surgery, due to topological obstructions; a major difficulty in the Perelman program is to show that such scenarios in fact cannot occur in a Ricci flow.)
In order to analyse these singularities, Hamilton and then Perelman employed the standard nonlinear PDE technique of “blowing up” the singularity using the scaling symmetry, and then exploiting as much “compactness” as is available in order to extract an “asymptotic profile” of that singularity from a sequence of such blowups, which had better properties than the original Ricci flow. [The PDE notion of a blowing up a solution around a singularity, by the way, is vaguely analogous to the algebraic geometry notion of blowing up a variety around a singularity, though the two notions are certainly not identical.] A sufficiently good classification of all the possible asymptotic profiles will, in principle, lead to enough structural properties on general singularities to Ricci flow that one can see how to perform surgery in a manner which controls both the geometry and the topology.
However, in order to carry out this program it is necessary to obtain geometric control on the Ricci flow which does not deteriorate when one blows up the solution; in the jargon of nonlinear PDE, we need to obtain bounds on some quantity which is both coercive (it bounds the geometry) and either critical (it is essentially invariant under rescaling) or subcritical (it becomes more powerful when one blows up the solution) with respect to the scaling symmetry. The discovery of controlled quantities for Ricci flow which were simultaneously coercive and critical was Perelman’s first major breakthrough in the subject (previously known controlled quantities were either supercritical or only partially coercive); it made it possible, at least in principle, to analyse general singularities of Ricci flow and thus to begin the surgery program discussed above. (In contrast, the main reason why questions such as Navier-Stokes global regularity are so difficult is that no controlled quantity which is both coercive and critical or subcritical is known.) The mere existence of such a quantity does not by any means establish global existence of Ricci flow with surgery immediately, but it does give one a non-trivial starting point from which one can hope to make progress.
It is always dangerous to venture an opinion as to why a problem is hard (cf. Clarke’s first law), but I’m going to stick my neck out on this one, because (a) it seems that there has been a lot of effort expended on this problem recently, sometimes perhaps without full awareness of the main difficulties, and (b) I would love to be proved wrong on this opinion :-) .
The global regularity problem for Navier-Stokes is of course a Clay Millennium Prize problem and it would be redundant to describe it again here. I will note, however, that it asks for existence of global smooth solutions to a Cauchy problem for a nonlinear PDE. There are countless other global regularity results of this type for many (but certainly not all) other nonlinear PDE; for instance, global regularity is known for Navier-Stokes in two spatial dimensions rather than three (this result essentially dates all the way back to Leray’s thesis in 1933!). Why is the three-dimensional Navier-Stokes global regularity problem considered so hard, when global regularity for so many other equations is easy, or at least achievable?
(For this post, I am only considering the global regularity problem for Navier-Stokes, from a purely mathematical viewpoint, and in the precise formulation given by the Clay Institute; I will not discuss at all the question as to what implications a rigorous solution (either positive or negative) to this problem would have for physics, computational fluid dynamics, or other disciplines, as these are beyond my area of expertise. But if anyone qualified in these fields wants to make a comment along these lines, by all means do so.)