1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

378 7. THE REDUCED DISTANCE

(i) The inequality

of 2 n

or -.6.P + IV Pl - R + 2r 2:: 0

holds in the weak sense on M x (0, T), i.e.,

(7.146)

1

72

r [ve. \l<p + (of + IVPl^2 - R + !!__) <p] dμdr 2:: 0

-r 1 }M or 2r

for any nonnegative C^2 function <p on M x [ri, r2] with 0 < r1 <

r 2 < T such that <p (·, r) has compact support for each r E [r1, r2] ·

(ii) For each r E (0, T) ,

(7.147)

P-n

2.6.P - IVPl^2 + R + --:::; O

r

holds in the weak sense on M x { r} , i.e.,

JM [-2\7£ · \l<p + <p (-1VPl

2

+ R + p ~ n)] dμ (r) :::; 0

for any nonnegative C^2 function <p on M with compact support.

Let rp be any nonnegative locally Lipschitz function which satisfies the

decay conditions:

rp (q, r), IV'P (q, r)I :::; ~e -cd;(o)(p,q)

c

for some constant c > 0. We now show that (7.146) and (7.147). hold for

<p = rp. In the proof of the main theorem in §11.2 of [297] (see Chapter 8 of

this volume), we need this generalization to be able to take <p = e-£. To see

this generalization, by the Bishop-Gromov volume comparison theorem, we

have the volume growth bound:

Volg(-r) Bg(-r) (p, r) :::; Geer

for some constant C > 0. By Lemmas 7.59 and 7.60 we know that both

integrals in (7.146) and (7.147) are finite. Let {<Pa} be a partition of unity on
M with compact support. Because of the finiteness of the integrals it suffices
to show that for each a the two inequalities hold for <p = <;bar{;, which is
Lipschitz and has compact support in space. Since <Pa'P can be approximated
by smooth functions along with its first derivative outside a set of arbitrary
small measure, by a proof similar to that of Lemma 7.113 we conclude that
for any a, (7.146) and (7.147) hold for <p = <Pa'P· Hence we have proved
that (7.146) and (7.147) hold for any nonnegative locally Lipschitz function

rp which satisfies the decay conditions rp (q, r), IV'P (q, r)I :::; c-^1 e-cd;caiCP,q)

for some constant c > 0. Note by Lemmas 7.59 and 7.60 that we can choose

<p = e-£ in (7.146) and (7.147).

Similarly we may take <p = e-£ in Proposition 7.128(iii) and we obtain
the following.