1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

118 19. GEOMETRIC PROPERTIES OF 1"-SOLUTIONS

such that the sequence { (M^3 , g 7 i ( B), qi)} converges to a nonfl.at 3-dimensional

gradient shrinking Ricci soliton

for 0 E (0, oo).

The limit solution (M 00 ,fJ 00 (-B)), BE (-oo,O) (with time reversed), is

a 3-dimensional t;;-solution with Harnack. Therefore, by Proposition 19.44,

(M 00 ,g 00 (-B)) is a t;;-solution, i.e., the curvature of § 00 (-B) is bounded.

We claim that the limit manifold M 00 is noncompact. Suppose that M 00

is compact. Then (M 00 , § 00 (8)) is a compact nonfl.at shrinking gradient Ricci

soliton with bounded curvature. By Perelman's classification theorem,^30

( M 00 , § 00 ( B)) is a compact quotient of 52 x JR or 53. On the other hand,

compact quotient solutions of 52 x JR may be ruled out since they are not

t;;-noncollapsed on all scales. Since a backwards limit of (M^3 , g (t)) is a com-

pact quotient of the shrinking round 53 , this implies that (M^3 , g (t)) itself

is actually a compact quotient solution of the shrinking round 53 (this fol-

lows from Hamilton's convergence theorem for solutions with positive Ricci

curvature on closed 3-manifolds; for example, one may use his 'curvature

pinching improves' estimate^31 ). However, such a compact quotient solution

of the shrinking round 53 is ruled out by the hypotheses of the theorem.

Now we may apply Theorem 19.55 to conclude that (M 00 , § 00 (-B)) is a

shrinking round cylinder 52 x JR^1 or its Z 2 -quotient.

STEP 2. Universal lower bound for the reduced volume and ""l -noncollap-

sing.

Choosing E = 160 (and A = 2 say) in Lemma 19.47, there exists a

universal constant 5 E (0, 1) such that

(19.69) £ (q, Ti) :S 0-l, TiRg (q, Ti) :S 5-l

for all q E BfJ(Ti) (qi, 10.JTi). Hence R9,,i (q, 1) :S 5-^1 for q E B9ri(l) (qi, 10)

and thus

R9 00 (q,1) :S 5-l for q E B9 00 (1) (q 00 , 10).

Since § 00 (1) is a round cylinder metric on 52 (p) x JR or its Z 2 -quotient,
where p denotes the radius of S^2 (p), we obtain

p ~JU.

Thus, if q 00 ~ (x 00 , f) 00 ) in the universal cover of M 00 is a lift of q 00 ,
then we have

(19.70) Bs2(p) ( xoo, JU) x Bm:. (iJoo, TJ2) c B9 00 (1)(qoo, 10) c 52 (p) x JR

(^30) See Lemma 1.2 in [152] (or Theorem 9.79 in [45]).
(^31) See Theorem 6.30 in Volume One.