22 27. NONCOMPACT GRADIENT RICCI SOLITONS
COROLLARY 27.36 (Volume growth of shrinkers with R :'.'.: o :'.'.: 0). If Q =
(Mn, g , f, -1) is a complete noncompact normalized shrinking GRS with R :'.'.: o :'.'.: 0
and 6 EM, then
(27.93)
VolB - n^28
0 (r):Sconst(Q,O)(l+r) -.
PROOF. By hypothesis, N (c) = c~(~) :'.'.: %· Hence, combining this inequality
with (27.91), for c :'.'.: n + 2 we have
P(c) ::; P(n + 2) e-!.:'+2(^1 - n2";,2
Hdc::; (n + 2)^8 e~ P(n + 2)c-^8.
From (27.85), we have P(c) :'.'.: 2 ~~~~ for c :'.'.: n + 2 and we conclude that
(27.94) V (c)::; 2 (n + 2)^8 e~ P(n + 2)c~-^8.
By (27.42) we obtain B 0 (2JC - C) C {! < c}. Hence, using (27.94), we derive
that
VolB 0 (2JC-C)::; 2 (n + 2)^8 e~ P(n + 2)c~-^8
holds for c :'.'.: n + 2 almost everywhere, so that
8 Ii (r + c )n-28
VolB 0 (r):S2(n+2) e2P(n+2) -
2
-
for all r ;:::: 2-)n + 2 - C. This completes the proof of Corollary 27.36. 0
REMARK 27.37. For 2 :S k :S n , we have the cylinder shrinkers (Mn,g)
(Nk,h) x JR_n- k, where Rc1i =~hand hence R 9 = ~· Since N must be compact,
we have VolB 0 (r) ~ const (1 + rt-k.
We may ask the following:QUESTION 27. 38. Can one show that if a simply-connected noncompact shrinker
(Mn, g, f , -1) satisfies R :'.'.: o > 0, then it must have a compact factor (Nk, h) with
k ;:::: min { m E IZ : m :'.'.: 25}?
QUESTION 27 .39. Can one show that if a complete noncompact shrinking GRS
(Mn,g,f,-l) satisfies AVR(g) > 0, then ASCR(g) < oo?
QUESTION 27.40. Can one show that if a simply-connected noncompact shrinker
(Mn, g, f, -1) does not have an JR. factor, then g has Euclidean volume growth?
4.4. Bakry-Emery volume comparison.
The analytic essence of the Bishop volume comparison theorem is the Bochner
formula
(27.95)1
2.0.l\7ul^2 = j\7^2 ul^2 +('Vu, \7.0.u) + Rc('Vu, 'Vu)
applied to the distance function. We now consider t he Bakry-Emery volume com-
parison theorem, which was used to derive (27.88) above, from the same point of
view.
Let (Mn, g) be a complete Riemannian manifold and let f : M -t JR. be a
smooth function. Adding (27.95) and
1
-2(\7 f , \7j\7ul^2 ) = -('Vu, \7 (\7 f , 'Vu)) + \7^2 f ('Vu, 'Vu)