1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

66 28. SPECIAL ANCIENT SOLUTIONS

By what amounts to the maximum principle, the function

Gmax (t) ~max G (x, t)

xEM

satisfies the om

(28.81)

for all t, in the sense of the lim sup of forward difference quotients. We conclude

that Gmax (t) = 0 and hence that (M^3 ,g(t)) is isometric to a spherical space form

S^3 /r. Indeed, to see this, we note that the solution to the comparison ODE

dr,,, i+;

dt (t) = -2cr,,, (t),

lim r C< ( t) = 00

t',.C<

is r,,, (t) = (2 (t - a))-e. By the ODE comparison principle, we then have Gmax (t) :::;

(2 (t - a))-e for all t >a and a> -oo. Hence for all t E (-oo, O] we conclude that

Gmax (t) :S lim (2 (t - a))-e = 0

a-;-oo

as desired. 0

5. Notes and commentary

§1. Theorem 28.l is due to Bing-Long Chen (see Corollary 2.3(i) in [61] and
also Proposition 5.5 in Huai-Dong Cao, B.-L. Chen, and Xi-Ping Zhu [45]); our
presentation of the proof follows more closely Takumi Yokota [450].
Corollary 28.2 is Corollary 2.5 in [61].
Theorem 28 .5 is Corollary 2.4 in [61].
See B.-L. Chen, Guoyi Xu, and Zhuhong Zhang [62] for a localization of the
Hamilton- Ivey estimate.
§2. For Theorem 28.11, see the original Zhenlei Zhang [452] or the expository
Theorem 17.13 in Part III.
Besides Natasa Sesum [365] and Zhou Zhang [454], results which support
Optimistic Conjecture 28.15 have appeared in a number of recent works, including
Bing Wang [429], Nam Le and Sesum [184], Joerg Enders, Reto Muller, and Peter
Topping [104] (the latter two references are discussed in the next chapter), and one
of the authors (see Theorem 6.40 of [77]).
For Theorem 28.21, see the original §12.l of Perelman [312] or the expository
Theorem 52.7 in Bruce Kleiner and John Lott [161].
Theorem 28.24 is Proposition 11.2 in Perelman [312]; see also the expository
Theorem 19.53 in Part III.
Theorem 28.23 is due to Simon Brendle [32].
For Theorem 28.30 see [76].
§3. The complete classification of 2-dimensional ancient solutions to the Ricci
flow with bounded curvature is due to the combined works of Daskalopoulos and
Sesum [94] (compact case), Daskalopoulos, Hamilton, and Sesum [93] (noncompact