1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

6. FIRST PROPERTIES OF THE BACKWARD LIMIT v 00 81

Since J 1 (g(t)) E [-C,O) fort E (-oo,-1] and since

-d d J1 (g (t)) =^1 ( -2R^2 - --R IY'vl^12 ) dμ52::; -2^1 R^2 dμ 5 2,

t 52 v 52

we obtain J~~ J5 2 R^2 dμ 5 2dt::; C. Since at each point x E 52 we h ave that R (x, t)

monotonically decreases to R 00 (x) as t ----t -oo, we co nclude that J 52 R'?x,dμ 52 = 0,

which yields R 00 = 0 a .e. on 52.

6. First properties of the backward limit v 00

Let (5^2 ,g(t)), t E (-oo,O), be a n ancient solution to the Ricci flow on a

maximal time interval. By (29.25), t he backward limit of the pressure function

(29.46) V 00 ~ lim V

l-'t- 00

exists and is a bounded nonnegative upper semicontinuous function on 52. The

understanding of v 00 is a crucial part of the proof of the main t heorem.

In the case of a round 2-sphere shrinking to a point at t = 0, we h ave v (t) = 2 ~tl,

t E ( -oo, 0) , and we have v 00 = 0. We sh all show in Proposition 29.16 below that

this property characterizes the round shrinking 2-sphere.
On t he other hand, for the King- Rosenau solution , t he C^00 backward limit of
VKR('l/J, 8, t) is

(29.47)

This corresponds to a backward Cheeger- Gromov limit being a fiat cylinder, where
the diffeomorphisms are identity maps. By (29.11), t he correspond ing limit on ~^2
is

(29.48)

Equivalently, v~R (s, e) ~ limH-oo v~R (s, e, t) =μfrom (29.10).

In this section, using the estimates proved in the previous two sections , we
show t hat _1.__gVoo 52 is a fiat metric in a weak sense. In §9 we sh all classify v 00. Unless
otherwise indicated , all of t he norms and inner products in this section are wit h
respect to g52.
By the estimates in §4 and by the Arzela- Ascoli theorem , we h ave the following.
LEMMA 29.12 (Convergence of v(t) to Voo)· The limit Voo is contained in C^1 ·°'
for all a E (0, 1) and has the properties that as t ----t - oo,

(1) v(t) converges to V 00 in C^1 •°',

(2) v^2 (t) converges to v'?x, in C^3 •°'; hence v (t) converges uniformly to v 00 in

C^3 • °' On Compact subsets of the open subs et Sl = { X : Voo ( X) > 0},

(3) IV'v( t) 12 converges to IY'voo 12 in C^1 ·°',

( 4) V (t) converges to V 00 weakly in W^2 •^2.

Since

l'V(vR 9 )1::; R 9 IY'vl + v IV' R 91 ::; R 9 IY'vl + v^112 l\7 R 9 l 9 ::; C ,

we h ave that ( v R 9 ) ( t) converges in C°' as t ----t -oo. Since v is bounded and R 00 = 0

a.e., we conclude that (vR 9 ) (t) converges to 0 in C°'. By (29.38), v 6.52V converges
in C°' to a function w 00 E C°' as t ----t - oo.