1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

5. EXISTENCE OF AN ASYMPTOTIC SHRINKER 109

From this and (19.48) we conclude that the second pair of terms on the

RHS of (19.47) is bounded by

for\ Vadg(T) (ti (T) ,12 (T)), ~~ (T)) dT

:::; for I ~~ I ( T) dT

:::; 4J37i/2,ei/2 (qa,f).

Thus we have

(19.51) dg(r) (qi,q2):::; for (:Tdg(T)) (ti (T) ,12 (T)) dT

• 4J37i/2 (.ei/2 (qi, f) + ,el/2 (q2, f)).

Now we bound the first term on the RHS of (19.51). The idea is to use

estimate (18.15)-(18.16) for the time derivative of the distance function;

namely, if for some r ( T) and K ( T)

(19.52) Re:::; (n -1) K (T) in B (ti(T), r (T)) U B(/2 (T), r (T)),

then

(19.53)

we shall chooser (T) and K (T) below, so that (19.52) holds.
Since

I I

1v.e1 J3

\;?Vf (q, T) =

2

Vf (q, T) :::;

2

yT,

by applying the fundamental theorem of calculus along geodesics with re-
spect tog (T), we have for any point q E Bg(T) (ta (T), r (T))

V3

Vi (q, T) :::; Vi (ta (T) 'T) +2y'Tr (T)

(19.54) :::; (f)i/4 ; Vf (qa, f) + V3

2

y'Tr (T)

by (19.50). Since (19.46) is symmetric in qi and q2, without loss of generality
we may assume that

. .e (qi, f) :::; .e (q2,T).

From Re ~ 0, (19.49), and this, we then have

Rc(q,T):::; R(q,T)

3

:::; -£ (q, T)

T

3 ((f)l/4 V3 )

2

:::; ; ; Vf(q2,f) + 2y'Tr(T) ..;_ (n-l)K(T)