1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

4. APPLICATIONS OF HAMILTON'S COMPACTNESS THEOREM 147

REMARK 3.32. In the above proof, we could have replaced Hamilton's

isoperirnetric estimate by Perelman's no local collapsing theorem in Chapter

6, which enables the application of the compactness theorem and at the same

time rules out the formation of the cigar soliton singularity model.

Now we give a proof that Type I limits are round 2-spheres using Perel-

man's entropy (see Chapter 6 for properties used in the proof below; the

reader may wish to come back to this part after reading that chapter).

PROOF #IIB, USING PERELMAN'S ENTROPY FOR LAST STEP. For the

last step in the above proof, we may use Perelman's entropy instead of

Hamilton's entropy. This has the advantage that the entropy is defined

for solutions with curvature changing sign, so that we may apply it to the

original solution g (t), t E [O, T), rather than the limit solution. We as-

sume that g (t) forms a Type I singularity. Let W(g (t), f (t), T (t)) denote

the entropy with T ~ T - t, which is defined for T E (0, T]. Taking f to

be the constant Ji (t) ~ -log Vol 9 ~7)(M), so that it satisfies the constraint

JM(4n'T)-^1 e-fidμ = 1, we see that

41fT

μ (g (t), T (t)):::; W(g, Ji, T):::; TRmax (t) - log Volg(t) (M) - 2.

In particular, by the long-time existence theorem for the Ricci fl.ow on the

2-sphere, we have Volg(t) (M) = 8n (T-t). Hence we have an upper bound

for the μ-invariant:

μ (g ( t) , T ( t)) :::; C - 2 + log 2,

where we have used the Type I assumption (T - t) Rmax (t) :::; C. In partic-

ular, by the monotonicity ofμ, ftμ (g (t), T (t)) ~ 0, the limit

μT ~ lim μ (g ( t) , T ( t))

t--+T

exists. Dilate the solution about (xk, tk) with 9k (t) = Rkg (tk + R7;1t) as

in the proof of Theorem 3.30. By the scaling property ofμ we have

μ (gk(t), Rk (T - tk) - t) = μ (g (tk + R;;^1 t) , T - tk - R'i/t).

Thus for each t E (-oo, w 00 ), the (maximal) time interval of existence of the
limit solution (M~,g 00 (t)),

μT = lim μ (gk(t), Rk(T - tk) - t) = μ (g 00 (t), W 00 - t).

k--+oo
(We may assume Rk (T - tk) -+ w 00 converges. Here we also used the con-
tinuity of μ (g, T) in g and the fact that the convergence, after the pull-back
by diffeomorphisms, of 9k (t) to g 00 (t) is globally pointwise in 000 since
M~ ~ M is compact.) Now the theorem follows from the result that a
solution having constant μ is a gradient shrinker. D