l. BASIC PROPERTIES OF GRADIENT RICCI SOLITONS 5for any unit speed minimal geodesic 'Y : [O, r (xe)] ---+ M joining 6 to Xe and any
co nt inuous piecewise C^00 function ( : [O, r (xe)] ---+ [O, 1] satisfying ( (0) = 0 and
((r(xe)) = 1, where 'Y'(s) ~ ~;(s).
We a lso have(27.19)
r<xc) d
-(\lf,\lr)(xe) =-(\i'f(O),'Y'(O)) - lo ds (\lf,\lr)dsr<xc)
= - (\i'f(O),'Y'(O)) - lo \1^2 f('Y','Y')ds
since \lr = "f^1 (s) and '7-y'(s)\lr = 0.
Therefore applying (27.1) to (27.19) and co mbining with (27.18), we haver<xc) 2 - €
6.1r(xe) :'S: (n -1) lo ((') ds - (\1 f(O), "f
1
(0)) + 2r (xe)r <xc)
+lo (1 - (^2 ) Re ('Y', 'Y') ds.Let ( ( s) = s for 0 :::; s :::; 1 and let ( ( s) = 1 for 1 < s :::; r (Xe). Then
- € 2
6.1r(xe) :'S: n - 1 - (\1 f(O), "f^1 (0)) + -r (x e) + - !fiax Re+,
2 3 B 0 (1)
(27.20)
where max.a 6 (1) Re+ ~ maxvETxM, IVl=l, xEBa(l) {Re (V, V), O}.
Applying (27.20) and (27.11) to (27.15) yields for all x E B a (c)
2 2
(27.21) -R (x) 2:: -e (x e)
n n
rJ'("(xc)) ( - 2 )2:: e n - l - (\lf(O),'Y' (0)) + -
3
!fiax Re+
C B 0 (1)+c(17'(~)r(xe) _
17
(r(xe)))-const,
2c c c^2where co nst < oo is independent of b and c.
Since Xe E Ba (( 1 + b) c) - Ba (c) in Case (ii), we h ave
(27.22) 1 < r (xe) < 1 + b.
- c
We now co nsider the nonexpanding and expanding cases separately, where we
primarily need to deal with Case (ii).
STEP 3. Proof of the theorem when€= 0 or -1. Here, Case (ii) must a lways
hold since otherwise (27.16) contradicts assumption (27.14). We take b = 2. Then
t he inequality (27.21) and - const:::; 171 :::; 0 imply that for all x E Ba (c)
(27.23) -R(x)^2 2:: ---const ( n - l + j\i'fl (0) - +^2 - !fiax Re+ ) - --const ,
n c 3 B 0 (1) c^2where co nst is independent of c. By taking c---+ oo, we conclude t hat R (x) 2:: 0 for
all x EM. This completes the proof of the t heorem in the nonexpanding case.