1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

1. COMPACT FINITE TIME SINGULARITY MODELS ARE SHRINKERS 11

We claim that

(17.32) A (g 00 (t)) > 0

for all t E (-oo, O].
To prove this, suppose by contradiction that >. (g 00 ( t')) = 0 for some
t' E (-oo, O]. Then by the >.-monotonicity formula,

A (g 00 (t)) = 0

for all t E (-oo, t']. This implies that g 00 (t) is a steady gradient Ricci

soliton fort E (-oo, t'] (see Lemma 5.28 in Part I). Since M 00 is compact,

g 00 (t) is Ricci flat for t E (-oo, O] (see Proposition 1.13 in Part I; we also
use uniqueness to extend to the whole time interval). This contradicts the
claim that any compact finite time singularity model cannot be Ricci fiat.
To see this claim, recall that (see Theorem 6.74 in Part I) any finite time

singularity model is ,,;-noncollapsed at all scales for some ,,; > 0 in the sense

that if B 900 (t)(x,r), r E (O,oo), is a metric ball such that

R 900 (t) :S: r-^2 for ally E B 900 (t)(x,r),

then

Vol 900 (t) B(x, r) :2': ,,;rn.

Since Rc 900 (t) = 0 on M 00 , this implies

Vol 900 (t) (M 00 ) :2': Krn

for all r E (0, oo) and t E (-oo, OJ, which is a contradiction since the LHS is
finite. This completes the proof of (17.32).

STEP 3. g 00 (t) is a shrinker. Now by (17.29) we have (we justify the
first equality in Lemma 17.14 below)

v (goo (t)) = ,lim V ('Pi (Qig (ti+ Qi^1 t)))

i-+oo

= .lim v (g (ti+ Q;^1 t))

i-+oo

= lim v (g (t))

t-+T

=VT

for all t E (-oo, OJ, where we used the fact that v(cg) = v(g) for any

c E (0, oo). Since v (g 00 (t)) is identically a constant and>. (g 00 (t)) > 0, by

the equality case of the v-monotonicity formula (see Lemma 6.35(2) in Part
I), we conclude that g 00 (t) is a shrinking gradient Ricci soliton. D

The following fact is used in the proof above.

LEMMA 17.14. If M~ is a closed manifold and gi -+ goo pointwise in

000 on M 00 , where >. (g 00 ) > 0, then

(17.33) v (g 00 ) = ,lim V (gi).

i-+oo