56 28. SPECIA L ANCIENT SOLUTIONS
Since R > 0, distances are decreasing under the Ricci fl.ow. Hence, if p E M -
Ba(C" +c^1 /^2 , t) , then
Bp(c -^112 ,t) C Bp(c-^112 , t) CM - Bo(C" , t) for all t E (-oo, t].
In particular, for such p we have
O<R(x,l)~c forxEBp(c-^1 /^2 ,t-c -^1 ) and tE[t-c -^1 , t].
Now, from scaling by c Shi's local higher derivative estimates (see Theorem 14.14
in Part II), we obtain that for each j ;::: 1 there exists CJ < oo such that
DFinally, we consider the first boundary t erm in (28.47). By the definition
of Cheeger- Gromov convergence, there exists an exhaustion {Ui} : 1 of IR x 51
and embeddings CfJi : (Ui ,P= ) ---+ (M, pi) such that (Ui, cpi 9 (t)) subconverges in
c= uniformly on compact sets to (IR x 51 , 9=). Let (s=, B= ) b e the cylindrical
coordinates of P= E IR x 51 and define the curve "ti ~ CfJi ( { s=} x 51 ) to b e t he
push forward of the distance circle passing through P=. Note that Pi E "ti. For i
sufficiently large, { s=} x 51 c Ui and hence "ti is an embedded loop in M bounding
a disk n i.
LEMMA 28. 45 (Vanishing of the first boundary t erm in the limit). W e have(28.51) llan; v (R) dal ---+ 0 as i---+ oo,
where v is the unit outward normal to 8Di.PROOF. Fix 0 EM. We have that limi-+= d 9 (pi, 0) = oo and that L 9 (8Di ) ~
C since it is approximately equal to the length of { s=} x 51 with respect to 9=.
Therefore 8Di ---+ oo and
I
r I/ (R) dal ~ L(8Di ) sup IV RI ---+ 0 as i---+ 00
Jan; an;by (28.50) for j = 1. D
3.5. Vanishing of the second boundary term in the limit.
In this subsection we shall control the second boundary term in (28.47). To
do this, we shall show that the spatial infinity of 9(t) is pointwise asymptotically
cylindrical. In particular, we shall use a crucial estimate for the co nformal factor
of 9(t) relative to the cylinder metric (see (28.62) below). Let (x, y) : M ---+ IR^2 b e
global conformal coordinates as b efore and from now on set M = IR^2. On IR^2 - {O}
let
9 ~ 9(t) ~ u(t)9cy),w eh re 9 cyl = d xx22+dy2 +yi. S' mce Rey! = 0, we h ave R = -u --1 6.cyl ln u. A Thus ft 9 =
-R9 implies ftu = 6.cy l ln u and hence
(28.52) :t ln u = 6. 9 ln u = -R < 0.
First, for each time t we shall obtain a uniform lower bound for u (see (28.56)
b elow). We shall b egin with a weaker estimate obtained from bounding g(t) from
b elow by a hyperbolic cusp. Since the noncompactness of IR^2 makes the application