1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

326 7. THE REDUCED DISTANCE

Hence b..L(q 7 , r) ?:: 0. We have

d+h(-)<l. L(q7,r+s)-L(q7,r)

-d T _ imsup

T s-+O+ S

1

:::; im. sup L(q7,r+s)-L(q^7 ,r)8L( - -;::;- q )

7 , T.

s-+0+ S UT

Then using (7.84), which holds for the barrier function L, we have

at A

OT (q 7 , r) :::; 2n - b..L(q7, r) :::; 2n.

We have proved that d;'Th (r) :::; o when Lis not C^2 at (q 7 , r). Hence we have

proved that d; 7 h(r):::; 0 for all r E (0, T). By the monotonicity principle for
Lipschitz functions stated in §3 (Lemma 3.1) of [179], h(T) is nonincreasing.
(ii) This follows from (i) and

_lim h(r) = !im min L (q, r)

T-+0+ 'T-+0 qEM

:S !imL(p,r) = (d 9 (o) (p,p))

2

'T-+0

= 0.

D

6.3. The reduced distance function£. To get even better equations
than those in Lemma 7.45, we introduce the reduced distance function£.

DEFINITION 7.49. The reduced distance f is defined by

(7.87)

1 1 -

f(x,T) ~ r;;:.L(x,T) = -

4

L(x,T).

2yT T

Let 7 E (0, T) and let'/': [O, r] ----t M be a minimal £-geodesic from p to
q and let K = K ('!', r) be defined as in (7.75). By (7.79), (7.80), and (7.81),

we have at (q, r),

(7.88)

(7.89)

(7.90)

8f = -^1 -K - ~ R

or 273/^2 r + '

2 1 g

IV'fl = -R--K 73/2 +-7'

1 n

!:lf < - ---K 273/2 +- 27 -R.

From these equations (which involve the trace Harnack integral K), (7.86)
and Lemma 7.48, we easily deduce the following which do not involve K.