This means that you can merge those two cases if you add time-shifting in your equivalence relation (in addition to isometry and rescaling). I would say that your statement is more a classification of time-zero slices of -solutions than a classification of -solutions strictly speaking.

Well, all this is irrelevant to the Poincare conjecture anyway…

]]>Regarding Theorem 1, I am basing this on Theorem 9.93 of Morgan-Tian (though I am grouping the cases a little differently). As I understand it, a C-component and a doubly C-capped strong epsilon-tube are topologically the same, but geometrically rather different; in the former the sectional curvature is bounded above and below everywhere, whereas in the latter the sectional curvature can be very small in the epsilon-tube component of the manifold (in planes parallel to the direction of the tube, of course). Also, if we normalise the maximal curvature to be 1, then C-components need to have diameter comparable to 1, but doubly C-capped epsilon-tubes can have arbitrarily large diameter.

Perhaps the key difference here between Morgan-Tian and Perelman is that in Morgan-Tian one tries to classify the global structure of kappa-solutions, even though for applications it is only the local structure (i.e. the canonical neighbourhoods) which is of importance. Thus, for instance, if one takes a doubly C-capped strong epsilon-tube, the canonical neighbourhood of any given point in such an object would be either an epsilon-neck, epsilon-cap, or a closed manifold in Perelman’s notation, depending on whether the tube is long or short and whether the point lies in the middle of the manifold or near one of the ends.

[In Morgan-Tian, the relevant canonical neighbourhood theorem is Corollary 9.94; the definition of canonical neighbourhoods on page 232 (and reproduced here as Definition 2) has exactly four cases and matches up with the corresponding notion in Perelman’s second paper.]

]]>“shrinking round” before : this manifold

is flat, so it does not shrink by Ricci flow (and I wouldn’t call it “round”.)

A more serious point: I am surprised to see five cases in the classification of

-solutions (Theorem 1), since in Perelman there are only four.

What is exactly the difference between case 3 “C-component” and case 5

“doubly -capped strong -tube” ?

I mentioned a small misprint. There’s no closing bracket in Theorem 3 after .

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