1547671870-The_Ricci_Flow__Chow

2. TENSOR EQUATIONS

fort E [to, to+ o] when we choose o > 0 sufficiently small, depending on

Mi;i;[~,T] I :t^9 1.

The evolution of Ac is given by

a a a 9

ot Ac = ot a + cg + c[ o + ( t - to)] ot.

Since b.Ac = b.a, we have

a a 9

ot Ac 2: b.Ac + ,8 +cg+ c [o + (t - to)] ot,

which we rewrite as

(4.7a)

(4.7b)

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We first choose Oo > 0 depending on g (t) fort E [O, T] to be small enough
so that on Mn x [to, to+ 60], we have

a 1

at^9 2: - 400 g.

This implies in particular that

(4.8)

on Mn x [to, to + Oo]. Since ,8 is locally Lipschitz, there exists a constant K

depending on the bounds for a and g on Mn x [O, T] (but not on c) which

is large enough that

,8 (a, g, t) - ,8 (Ac, g, t) 2: -Kc [oo + (t - to)] g 2: - 2Kc6og

on Mn x [to, to+ Oo]. Then if we choose o E (0, Oo) so small that

1

o < 4K'

we have

(4.9)

Hence combining (4.8) and (4.9) to (4.7) shows that

a

( 4.10) ot Ac > b.Ac + ,8 (Ac, g, t)

on Mn x [to, to+ o]. We claim that Ac > 0 on Mn x [t 0 , to+ o]. Suppose the

claim is false. Then there exists a point and time ( X 1' t1) E Mn x (to' to + o]

and a nonzero vector v E Tx 1 Mn such that Ac > 0 for all times to :St< t1,

but