1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

20 27. NONCOMPACT GRADIENT RICCI SOLITONS

THEOREM 27.33 (The AVR exists and is bounded for shrinkers). Let (Mn,g,

f , -1) be a complete normalized noncompact shrinking GRS. Then AVR(g) exists

and is bounded above by a constant depending only on n.

PROOF. We modify the volume ratio V~) by defining the quantity

C 2

(27.82) P(c) ~ V ~c) - ~~~.

c2 C2

For convenience, let N (c) ~ }~>(l). Note that ~~~~ is the average scalar curvature

over the set{!< c}. Using the ODE (27.8 1 ), we compute that

P'(c) = V' n(c) _ '!!:_ Vn (c) _ R: (c) + n + 2 ~ (c)

(^27 ·^83 ) c2 2 c2+1 c2+1 2 c2+^2

• 
  

  -(-n+2)R(c)

  
  


• 
  

  (^1 2) C c2 '1+ 1
  (1-n~^2 ) N (c)
  =- 1-N(c) P(c).
  By (27.79), we have monotonicity, i.e.,
  (27.84) P'(c):::; 0
  for c 2'. nt^2 and we have

  
  

(27.85) (^1 - .7:1:) V~) :::; P(c):::; V~).

2c c 2 c 2

Hence, by (27.73) and (27.85), we have

(27.86)

. V(c)
2nwn AVR(g) = hm -- lim P(c) ,
c --+oo cn/2 c--+oo

where the limits exist by (27.84).
Finally, we show that AVR(g) is bounded above by a constant depending only
on n. We have for all c 2'. nt^2 ,

P ( C ) < - P (n + 2) < V(~) < VolB^0 h /2(n + 2) + ¥n).

2 - ( nt2) ¥ - ( nt2) ¥ '

the last inequality follows from (27.47). Since AVR(g) is independent of the base-
point, we may choose 0 to be a minimum point 0 off. The desired result follows
from

Claim. There exists C(n) < oo such that

Vol Bo ( J2(n + 2) +

3

8

5

n) :::; C(n).

Hence, by (27.86) we have AVR(g) :S wn( 2 ~~~))"/2 ·

Proof of th e claim. Let an= J2(n + 2) +^3 ; n. By (27.44), for x E B 0 (an) we

have

(27.87)

1 1

i'Vfl (x) :S Vf(x ) :S 2(V2ri:+d(x,O)) :S 2(V2ri:+an) :S 3.4n.