1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

5. NOTES AND COMMENTARY 169

Note the following:

(a) A steady GRS may not necessarily be 11:-noncollapsed for any 11: > 0. For

example, we may take the product of the cigar and l.Rn-^2.

(b) An expanding Ricci soliton may be neither gradient nor 11:-noncollapsed

for any 11: > 0 (see Lott [208]).

5. Notes and commentary
   §1. For Definition 30.1, see Definition 3 .4 in Enders [103].
   Theorem 30.10 is Theorem 3.3.1 of Enders [102] in the Type A case; see Naber
   [273] for the Type I case.
   Regarding the Type A condition, we remark that matched asymptotics calcu-

lations by Angenent and two of the authors [11] indicate that for any n , k ~ 3

there should exist rotationally symmetric degenerate neckpinches with space-time
sn x [O, T) and curvature blow-up rate given by

sup IRml ( · , t) ,...., C (T - t)-^2 +t.

sn

Note that when k = 3, such a solution is Type A.

§2. Theorem 30.13 is in Naber [273] and is Corollary 4.3 in Enders [103]
(there it is stated more generally for Type A solutions). See also the antecedent
Proposition 11.2 in Perelman [312] (or the expository Theorem 19.53 in Part III).
For Theorems 30.22 and 30.27, see Enders, Muller, and Topping [104]. For
related work, see Le and Sesum [184].