1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

6. IMPROVED VERSION OF NLC AND DIAMETER CONTROL 267

Hence

VolB(p,s) > (~)nVolB(p,s/2)

sn - 2 (s/2t

> (~)nk VolB (p,s/2k)

- 2 (s/2kr

;::: (~) nk e_3n36eμ(g,s2;22k)e-3nMR(p,s/2k)

;::: (~) nk e_3n36evr(§)e-3nMR(p,r).

The theorem again follows in this case. D

Now we apply Proposition 6.72 to solutions of the Ricci fl.ow and obtain
the following improvement of the no local collapsing theorem.

THEOREM 6.74 (No local collapsing theorem improved). Let (Mn, g (t)),

t E [O, T), be a solution to the Ricci flow on a closed manifold with T < oo

and let p E (O,oo). There exists a constant K, = K,(n,g(O),T,p) > 0 such

that if p EM, t E [O, T), and r E (0, p] are such that

then

for all 0 < s :Sr.

R :S r-^2 in Bg(t) (p, r),

Volg(t) Bg(t) (p, s)

------>K, sn -

PROOF. By (6.61) and the definition of Vri we have

Vr (g (t)) ;::: V yP2+i' (g (0))

for r E (0, p] and t E [O, T). Then the theorem follows from Proposition
6.72. D

6.2. Diameter control. In this subsection we show how ideas related
to the previous subsection can be used to obtain a diameter bound for solu-
tions of the Ricci fl.ow in terms of the L(n-l)/^2 -norm of the scalar curvature.
This result is due to Topping [357] and our presentation essentially follows

his ideas. Recall that Proposition 6. 72 implies that if (JVt.n, g) is a closed

Riemannian manifold, then for any p E J\/t.n and 0 < s < oo, we have

(6.98) VolB sn (p, s) > - e_3n36ev(§)e-3nMR(p,r) '

using v (§) ~ inf 7 E[O,oo) μ (§, T) :S Vr (§) for all r. Recall that by Corollary
6.34, if the lowest eigenvalue is positive, i.e., A.(§) ~ A.1(-4b.,q + R_g) > 0,

then v (9) > -oo.