24 27. NONCOMPACT GRADIENT RICCI SOLITONS
since lim,._, 0 rn-l J 1 (r) = e-f(Ol. Integrating this yields the following (one deals
with the possible nonsmoothness of S(O,r) in t he same way as the Bishop volume
comparison theorem):
THEOREM 27.41 (Bakry-Emery volume comparison). Let (Mn,g) be a com-
plete Riemannian manifold and let f : M ---+ JR be a smooth function. If 6 E M
and r > 0 are such that Rc1;::: 0 and IVJI:::; A in B 0 (r) for some A> 0, then the
!-volume
(27.104)
Vol1 Bo(r) = r e-f dμ:::; nwn r e-f(O)+Afr n-ldr:::; Wne-f(O)+Arrn,
1B 6 (r) lo
where Wn is the volume of the unit Euclidean n-ball. In particular, if f :::; C in
B 0 (r), then(27.105)From the form of the estimates we see that Bakry-Emery volume co mpariso n
is more effective at bounded distances.
4.5. Euclidean volume growth via the Riccati equation.
There is the following essentially equivalent version of the volume growth bound
in Theorem 27.33. The proof we give is due to Munteanu and J. Wang.
THEOREM 27.42 (Shrinkers have at most Euclidean volume growth). Let
(Mn, g , f , -1) be a complete normalized noncompact shrinking GRS. Then for any
6 EM we have
(27.106)where Wn is the volume of the unit Euclidean n-ball. In particular, if 0 is a mini-
mum point off, then by (27.43) we have that VolBo(r):::; wne1J-rn for r > 0.PROOF. On a shrinker we have R c1 = ~g. By (27.101), we have(27.107) i_ ln ( J ( r) ) = H ( r) - n -
18r rn-l r
:::; - - + -r 8J (r) - -^21 r r-(r)dr. 8J
6 8r r^2 0 8rSince. l' lmr_,0 rn-1 J(r) =^1 , we h ave
(27.108) ln (!(f)) :::; r (-~+DJ (r) - 2- r /
31 (r)dr) dr
rn-l lo 6 or r^2 lo 8r
= - -f^2 - f(r) + j(O) +-:: 21r r -8f (r)dr,
12 r 0 orwhere we integrated by parts and used limr_, 0 ~ J; r%{:(r)dr = 0 to obtain the last
line. Note t hat j(O) = f (0). Another version of this formula is
(27.109) ( J(f)) r
(^2 2) r
ln fn-1 :::; - 12 + j(O) + f(r) - f lo f(r)dr.