This question has arised from the links of the comments in the question What are the fundamental problem in solving the NAVIER–STOKES equation.. As I understand Ricci flow has been used in solving the Poincaré conjecture, further it is also mentioned that "ideally or in principle" Ricci flow can be used to solve Naivier-Stokes problem as well. Both, Terrence Tao and Grisha Perelman has mentioned that such approaches to solve the Naivier-Stokes is not easy. Specifically Terrence Tao quote:
"Strategy 2 may have a little more hope; after all, the Poincaré conjecture became solvable (though still very far from trivial) after Perelman introduced a new globally controlled quantity for Ricci flow (the Perelman entropy) which turned out to be both coercive and critical. (See also my exposition of this topic.) But we are still not very good at discovering new globally controlled quantities; to quote Klainerman, “the discovery of any new bound, stronger than that provided by the energy, for general solutions of any of our basic physical equations would have the significance of a major event” (emphasis mine)"
I am trying to understand the problem of associating "globally controlled quantity" into the problem. What does it really mean? Does it also mean that we have no idea what the topology is? We know that Manifold are locally Euclidean but to solve Naivier-Stokes we need "globally controlled quantity" to construct the topology? Few comments would be appreciated.