1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

62 28. SPECIAL ANCIENT SOLUTIONS

where C 1 = e^40 I 71"^3 sup1R2 R. By (28.63) with E = 2 1 we have that ew - E L^2 (B11").

Now we may apply a standard interior elliptic estimate to (28.65) to obtain (see

Theorem 4.1 in [144])

ll(lnu)+llL=(B,,.;2) :::; Cll(ln u)+llu(B,,.) + Cll.6.cyl ln u llL2(B,,.) :::; c. D

Let j =i= - ln u. We now estimate the higher derivatives of J.

LEMMA 28 .53 (Higher derivative bounds for }). For any compact time interval

[t1, t2] C (-oo, 0) and k E N there exists a constant Ck < oo such that

k A 2 1

IY'cyiflcy1:::; ck on [2, oo) x 5 x [ti , t2l·

PROOF. By Lemmas 28 .48 and 28 .52, we have that for each compact time

interval [t 1 , t 2 ] C (-oo, 0) there exists a constant C < oo such that

(28.66) lfl :SC in [1, oo) x 51 x [t 1 - 1, t2]·

By (28.52), we have that gJ = .6. 9 } , where gtg = -Rg. Since Rand its covariant

derivatives are uniformly bounded on ([1, oo) x 51 ) x [t 1 - 1, t2], one can prove

uniform higher derivative bounds for J in the interior by the Bernstein technique

(compare with Section 3 of Chapter 14 in Part II). Namely, there exist constants

ck < 00 such that

(28.67)

For example, using (28.44), we compute that (the quantities are with respect

to g(t))

(28.68)

Using lfl :SC and defining P = (4C^2 + f2)1V'fl^2 , we compute that

(28.69) a; = .6.P - 2(4C^2 + }2)1\7^2 }1^2 - 21\7 }1^4 - 2\7(}2) · V'l\7 }1^2

< .6.P-_2p2

• 12 5C^4

We can localize (28.69) to obtain a uniform estimate for IV' JI on [2, oo) x 51 x [ti, t2] ·

Similarly, one can prove uniform bounds for the higher covariant derivatives of }(t)

with respect to g(t) on [2, oo) x 51 x [t 1 , t 2 ].

Now (28.66) and (28.67) imply that

k A 1

IY'cyiflcyl :SC in [2, oo) X 5 X [t1, t2]·

For example, fork= 1 we have that IY'cyiflcyl = u^112 l\7 9 fl 9 is bounded. For higher

derivatives, we compare V'~yl with \7;. D

By the derivative estimates for J = -ln u of Lemma 28.53 and by the Arzela-

Ascoli theorem, we can improve the Cheeger- Gromov convergence to pointwise
convergence; i.e., we may choose the diffeomorphisms to be translations in the s
direction of the cylinder. Recall that the cylindrical coordinates of the basepoints
Pi are (si, Bi), where Si -7 oo.