1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

6 27. NONCOMPACT GRADIENT RICCI SOLITONS

STEP 4. Proof of the theorem when c: = l. Define 'T/ : [O, oo) ~ [O, 1] so as to

satisfy the properties that 'T/ is C^00 on (0 , 1 + c),

(27.24)

and

(27.25)

'T/ ( ) u ::::::. { ~+e-u e

0

ifuE[0,1],

if u E [2, 1 + c),

ifuE [l+c, oo),

-~ ~ 'T)^1 ~ 0 and ITJ"I ~ const on (0, 1 + c),

c

where const < oo is independent of c. Note that 'T/ satisfies the prior conditions in

(27.11) with b = c. Moreover, for the purposes below, the nondifferentiability of 'T)

at u = 1 + c shall not be an issue since 'T/ (1 + c) = 0.

If Case (i) holds, i.e., X e E B 0 (c), then we have the estimate (27.17) in B 0 (c).
Now assume that we are in Case (ii). Then Xe E B 0 ((1 + c) c) - B 0 (c).
(ii)(a) If X e E B 0 ((1 + c) c)-B 0 (2c), then since 'T/ (u) = l+~-u and 'T)^1 (u) = -~
for u E [2, 1 + c), from (27.21) we have that for all x E B 0 (c)

(27.26) ~R (x) ::'.". -

1

2 (n -^1 +IV JI (0) + - 2

1

r (x e) + ~

3

!_Ilax: Re+)

n c B 0 (1)

1 + C - r(xc ) 1

---~e-- - const

c c^2

::'.". ---l+c - -1 ( n - 1 + IV f I ( 0) - + -^2 !_Ilax Re+ ) - -^1 const.

c c^2 3 B 0 (1) c^2

(ii)(b) Otherwise, if X e E B 0 (2c) - B 0 (c), then since 'T) ~ 1, from (27.21) and
from (27.25) we have that for all x E B 0 (c)

(27.27) -^2 R ( x) ::'.". --2( n - 1 + IV f I ( 0) - + c +^2 - !_Ilax: Re+ ) - 1 - -^1 const.

n c^2 3 B 0 (1) c^2

From combining the estimates (27.26) and (27.27) for Case (ii) with the esti-
mate (27.17) for Case (i), we have that for all x E B 0 (c)

(27.28) -R^2 (x) ::'.".min { ---l+c - -^1 const, --^2 (const +c) - 1 - -^1 const, -1 }.

n c & & &

We then conclude when c: = 1 that, from taking c ~ oo in (27.28), ~R (x) ::'.". -1 for
all x EM. This completes the proof of Theorem 27.2 in the expanding case. D

1.3. Characterizing completeness of GRS structures.

Recall that the vector field V f is complete if for each p E M, the integral

curve /p to V f with /p (0) = p may be defined on all of R In this case, V f

generates a 1-parameter group of diffeomorphisms { cpt} tEIR of M which is given by
'Pt (p) = /p (t) for any p EM and t ER We are interested in finding conditions to

ensure that the vector field V f is complete on a GRS.

First, by applying the lower bound for R in Theorem 27.2 to (27.6), we obtain
the following result.