32 27. NONCOMPACT GRADIENT RICCI SOLITONS
Therefore(27.144) 1 ;~~~ro R ('y(s)) ds::::; e4(n-1)(1+2v'J(O))ro.Combining this wit h (27.142), we have
(27.145)
r (x)
Jo Re ('y', 'Y
1
) ds::::; (n - 1) r 01 + C 1 ,where
(27.146)2 -
C1 = n - 1 + - max _ Re (V, V) + e^4 (n-l)(l+^2 v'7(0)).
3 VETyM, IVl=l, yEB 6 (1)Applying (27.145) to (27.141) while using r0-^1 = 4C~~)l) ~ 1, we obtain (27.140).
STEP 2. Applying the inequality (27.140) to prove the claim. Now define(27.147) r1 = inax {n-l - - }
2- , Ci + l'Vfl (0).
By (27.138) and (27.140) with x = u (u), we have that
1 - 1
(\7 f, 'Y^1 (r (u (u)))) ~ ~!7' (u (u)) - C1 - l'Vfl (0) ~ Sr (u (u))for any u E (-oo,O]. Here, 'Y: [O,r (u (u))] -t Mis a minimal unit speed geodesic
joining 6 to u (u). Since ~~ r (u (u)) ~ (\7 f , 'Y' (r (u (u)))), we obtain (27.139) for
all u E (-oo, 0). D
As a consequence, we have the following result, originally due to Carrillo and
one of the authors.
THEOREM 27. 54 (Ric~ 0 and Ric¢: 0 shrinker=;. AVR = 0). If (Mn, g , f , - 1)
is a complet e noncompact non-Ricci fiat shrinking GRS with nonnegative Riccicurvature, then A VR (g) = 0.
PROOF. This follows directly from a combination of Theorem 27.53 and Corol-
lary 27.36. D
REMARK 27.55 (Ricci flat manifolds with AVR > 0). There are many exam-
ples of 4-dimensional complete noncompact Ricci flat ma nifolds (hyper-Kahler and
asymptotically locally Euclidean) with AVR (g) > 0.
Note that Feldma n , Ilmanen, and one of the authors [111] h ave described
examples of complete noncompact Kahler shrinkers, which have AVR > 0 and have
Ricci curvature which is negative somewhere. R elated to R emark 27.37, we ask the
following:
PROBLEM 27.56. Is there an example of a noncompact shrinker with Re ~ 0
which does not split as the product of a compact shrinker and a Euclidean space?