1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

ioo 29. C OMPACT^2 -DIMENSIONAL ANCIENT SOLUTIONS

'' ' '

'' ' '

FIGURE 29.5.

5g_(t)(s x, t )

(^59) x, (t)(r t )

By (29.97), there exists St E (pt, rt) such that sg;t)(st) c .Bg(tl(k/2) and

(29.98)

Since (.Bg<t\k/2),g(t)) is a nearly Euclidean closed disk, the sets sg;t)(rt) and

sg;t)(st) are loops bounding disks Di and D 2 in .Bg<tl(k/2), respectively. We also

have that sg;t)(rt) u sg;t)(st) bounds an embedded a nnulus. This and (29.98)

. imply that D 2 contains Di. Hence q E D 2 as well as D2 n 8Bg(t)(k/2) -=f. 0. Since

Bg<t\k/2) is almost Euclidean , we conclude that Lg(t)(s g;t)(st)) 2 ~- We now

obtain a contradiction by choosing k sufficiently lar ge since

holds by (29.90). D

Using the proof of Proposition 29.30, we can obtain the following.

LEMMA 29.31 (Nonround solutions cannot have pla ne limits). Assume that

(5^2 , g (t)) is not a round shrinking 2-sphere. Suppose that ti --+ -oo and qi E 52

are such that (5^2 , g(ti + t), qi) converges in the pointed C^00 Cheeger- Gromov sense

to (M~,g 00 (t),q 00 ). Then (M~,g 00 (t)) cannot be the fiat plane.

PROOF. Suppose that (M~,g 00 (t)) is the flat p lane. The key point is that

the pointed Cheeger- Gromov convergence does not use rescaling of the sequence

of solutions g(t +ti)· By the proof of Proposit ion 29.30 we may assume that, for

ltl sufficiently large, 5 !ht) contains a large almost Euclidean b all. We may now

repeat the rest of the proof to obtain the lemma. D

9. Classifying the backward pointwise limit

Let g(t) = v(t)-ig 5 2, t E (-oo,O), b e a maximal solution of the Ricci fl.ow on

52. The main goal of this section is to prove Proposition 29.36 b elow. This result

classifies the specific form of the backward limit of v (t) either as vanishing or as

co rres ponding to a fl.at cylinder. The proof hinges on the monotonicity of certain