1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

ll6 19. GEOMETRIC PROPERTIES OF iv-SOLUTIONS

On the other hand, by (19.66) and the volume comparison theorem (again

Rm 9 ri ( e) 2:: 0), we have for all i

r ( 1) (47rtJ)-n/2 e_,e.9ri(q,B)dμgri(B) (q)

JM-B 9 ri(8) qri>e

:S 1 1 (47rtJ)-n/2e"'tze_c<;)d~ri(e)(q,qrJdμgri(e)(q)

M-Bgri (8) (qri'e)

(19.68) ::; r ( 47rtJ)-n/^2 e nt^2 e-c<;) lxl^2 dx

}JR.n-B(~)

= 6 (n,.svte),

where 6 ( n, cv'e) depends only on n and cv'e and where 6 ( n, cv'e) --+ 0 as

c--+ 0 (for fixed e > 0).

Combining (19.64), (19.67), and (19.68), we conclude that

_lim VahtJ) = Voo(O),

i-too ·

which is (19.63). This completes our discussion related to the proof of

Theorem 19.53.

EXERCISE 19.54.

(1) Let g (T) = 2 (n -1) (T + 1) gsn, TE [O, oo), where gsn is the stan-

dard metric on the unit n-sphere. Show that for tJ E (0, oo)

900(0) ~ Hm grJO) = 2 (n - 1) tJ gsn,

i-too

where the limit is pointwise in 000.

(2) Let (Mn,gi(tJ)), e E [O,oo), be Einstein solutions of the backward

Ricci fl.ow with Rgi(o) = ~Ti > 0. Assume 9i (1) pointwise converges

to a metric g 00 (1) on M. Show that

foo (q,e) ~ _lim f9i (q,e) = ~

i-too. 2

for all q EM and e E (0, oo). Hint: See subsection 7.1 of Chapter

7 in Part I for a formula for f, = f,9i.

6. The A;-gap theorem for 3-dimensional A;-solutions
   First we recall the following result of Perelman on the nonexistence of
   3-dimensional A;-noncollapsed gradient shrinking solitons (see the original
   §1.2 of Perelman's [153] or §9.6 of [45]).

THEOREM 19.55 (Nonexistence of 3-dimensi.onal compact shrinkers with

Rm> 0). There do not exist complete noncompact 3-dimensional gradient

shrinking solitons which have bounded positive sectional curvatures and are

A;-noncollapsed at all scales.