1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

268 6. ENTROPY AND NO LOCAL COLLAPSING

THEOREM 6. 75 (Topping). Let n 2: 3 and let (ktn, 9) be a closed Rie-

mannian manifold with v (9) > -oo. Then

diam (kt, 9) S max { ~~, 6e^3 n^37 e-v(§)} JM R:2

1

dμ,

where Wn is the volume of the unit n-ball in llln.

PROOF. Let <5 (9) ~ min { w 2 n, 11; (9) e_^3 n}, where 11; (9) ~ e_^3 n^36 e^11 (9).

N ate t h at 1 Ims---tO. VolB(p,s) 8 n = Wn an d 1 lms---too. VolB(p,s) 8 n =^0 Slnce · MA IS · · C^1 OSe d ·

Hence for any point p E kt, there exists s(p) > 0 such that Vol~~~(p)) =

<5 (9) and Vol~Jp,s) 2: <5 (9) for s E (0, s(p)].
Applying inequality (6.98), we have
(6.99) MR(P, s(p)) 2: 1.
This implies there exists s' (p) s s(p) such that

(s'(p))2 { R dμ > 1.

VolB(p, s'(p)) }B(p,s'(p)) + -

Applying the Holder inequality, we have

VolB(p, s'(p)) < f R d

(s'(p))^2 - JB(p,s'(p)) + μ

2

s ( { R:2

1

dμ)n-l [VolB(p,s'(p))]~=i,

j B(p,s'(p))

so that

VolB(p, s'(p)) s. { R:2^1 dμ.

( s' (p) )n-1 } B(p,s'(p))

We have proved that for every p EM, there exists s'(p) > 0 such that

<5(A) '() < VolB(p,s'(p)). '()

g s p - ( s' (p)) n s p

VolB(p, s'(p)) 1 nz-^1

= I n-1 S R+ dμ,

(s (p)) B(p,s'(p)) ..

where the first inequality follows from the definition of s(p) and s'(p) S s(p).

To finish the proof of the theorem, let '"'! be a minimal geodesic whose

length is the diameter of (kt, 9). One can show that there exists a countable

(possibly finite) number of points Pi E '"'(such that B (Pi, s' (Pi)) are disjoint

and cover at least 1/3 of'"'( (Vitali covering-type theorem). Then

t diam (kt, 9) SL 2s

1

(Pi) s 0

2

(A)L1 , R:2

1

dμ

i g i B(pi,s (Pi))

S <5 ~9) JM R:21 dμ.