1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

8 17. ENTROPY, μ-INVARIANT, AND FINITE TIME SINGULARITIES

On the other hand,

so that

JM 4 JV'wJ

2

dμ :S: JM (4 JV'wj

2

+ (R-Rmin (g)) w^2 ) dμ

= Y (g,w) -Rmin (g),

W (g, f, r) ~ ( r - i) g (g, w) - ~ log ( 4nr) - n - n log C 8 (g) + i Rmin (g).

Choosing f, to be the constrained minimizer of W (g, · , r), we have

μ (g, r) ~ (r - i) g (g, wr) - ~ log(4nr) - n - nlogC 8 (g) + iRmin (g),

where Wr ~ (4nr)-nl^4 e-f-,./^2. If r ~ ~' then we obtain (17.25). This
completes the proof of the lemma. D
1.3. Volume lower bound for Ricci flow solutions with A :S: 0.
As a geometric application of the upper and lower bounds for μ, we have
the following, which is Lemma 3.1 in [197].

LEMMA 17.8 (Lower bound for the volume of a solution when A :S: 0). If

(Mn, g ( t)), t E [O, T), is a solution to the Ricci flow on a closed manifold

with A (g ( t)) :S: 0 for all t E [ 0, T), then there exists c1, c2 E ( 0, oo) depending

only on g (0) such that

(17.28) Vol (g (t)) ~ cie-c^2 t

for all t E [O, T).

PROOF. By taking g = g (0) and r = ~+tin (17.25) and by Perelman's
μ-invariant monotonicity formula (see Lemma 6.26 in Part I), we have

μ (g(t), i) ~ μ (g(O), i +t)

~ t.A (g ( 0)) - ~ log ( 4n ( i + t))

n

- n - nlog Cs (g (0)) + SRmin (g (0)) ·

On the other hand, since A (g (t)) ::; 0, by (17.22) we have

μ (g (t), i) :S: log Vol (g (t)) - ~log (n

2

n) - n + 1.

Hence

log Vol (g (t)) ~ t.A (g (0)) - ~log (4n (i + t)) + c,

where

c =;=. -nlog C 8 (g (0)) + n SRmin (g (0)) + n 2 log (nn) 2 - 1.

We conclude that

Vol (g (t)) ~ ec ( 4n (i + t) )-n/

2

et.A(g(O)).

The lemma follows easily. D