1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

30 27. NONCOMPACT GRADIENT RICCI SOLITONS

and hence

(27.134) R(xo) 2 R(u(u)) for all u E (-oo,O].

Claim. For any x 0 EM - Ba(Sr 1 ) , we must have either Case (1):

(27.135) R (O' (u 1 )) 2 1 for some u1 E (-oo, O]

or Case (2):

(27.136) O' ( u2) E Ba (Sri) for some u2 E ( -oo, 0).

Now assume the claim. If Case (1) holds, then (27.134) implies

R(xo) 2 R(O'(u1)) 21.

On the other hand, if Case (2) holds, then (27.134) implies
(27.137) R(xo)2R(O'(u2))2'. giin R(x).
xEB 0 (8r1)
Since g is not Ricci fl.at, by Proposition 27.S(3) the RHS is positive. Therefore, in
either Case (1) or Case (2) we h ave

R (xo) 2 min {1, giin R (x)} ~ 6 > 0.

xEB 0 (8r1)

Since x 0 EM - Ba(Sr 1 ) is arbitrary, the theorem is proved, modulo the claim. 0

It remains for us to present the following:

PROOF OF THE CLAIM. Suppose that the claim is false. Then there exists xo E

M - Ba(Sr 1 ) such that

R(O'(u))<l

(27.13S)

for all u E ( -oo, O],

for all u E ( -oo, 0).

Let r ( ·) = d( · , 0). The whole aim of the proof is to show that

(27.139) ~~ r (O' (u)) 2 ~r (O' (u))

for all u E (-oo, 0) , where ~~ denotes the lim inf of backward difference quotients.
This will prove the claim since it contradicts O' (u) ~Ba (Sri) for all u E (-oo, 0).

STEP 1. For x EM - Ba (4 (n - 1)) with R (x):::; 1, we have

I 1 -

(27.140) (V f , 1 (r (x))) 2

4

r (x) - C1 - IV fl (0),

where 1: [O, r (x)] --t M is a minimal unit speed geodesic joining 0 to x and where
C 1 is given by (27.146) below.

Proof of (27.140). By integrating the shrinker equation Rc(1', 1 ')+(!01)" =

~,we have

(27.141)

1 r (x)

(\J f , 1

1

(r (x))) = 2r (x) - Jo Rc(r', 11 ) ds + (\J f, 11 (0)).

Regarding the second term on the RHS, the second variation formula implies (see
(27.49))
r(x) r (x)
Jo (^2 Re (r', 1
1
) ds:::; (n - 1) Jo ((')

2

ds