1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

304 7. THE REDUCED DISTANCE

Since 72 < T, by (7.27) and the Bernstein-Banda-Shi derivative estimate
(Theorem Vl-p. 224), there exists a constant C (n) < oo such that

(7.44) IVR(x,7)1:::; C(n)Co ~ C 2

-Jmin {T-72, C 01 }

(7.45)

Applying the bounds (7.27) and (7.44) on the curvature and its first deriv-
ative, we have

- d I -d/312 <2VrCo I -d/312 +TC2 I -d/31

dO' dO' g(a2/4) - dO' g(a2/4) dO' g(a2/4)

(7.46) < 3VTCo I d/312 + Ci T3/2.

• dO' g(a2; 4 ) 4Co

In view of the above ordinary differential inequality (7.46), given positive

constants c1 and c2, consider the ODE

dA

(7.47) dO' = c1A + c2.

Then for positive solutions A ( O') ,

log ( c1A + c2) ( O') = log ( c1A + c2) ( 0'1) + c1 ( O' - 0'1) ,

which implies

A(O') = ec1(a-a1)A(O'l) + ~~ (eci(a-a1) -1).

Hence, comparing the solution to the om (7.46) with (7.43) to the solution to

the ODE (7.47) with A ( 0'1) = IVl^2 and taking c1 = 3VTCo and c2 = ,i3 0 T^312 ,

we have

I

d/31

2

:::; e6Co-/T( .Ji-Fi) IVl2 + Ci~. (e6Co-/T( .Ji-Fi.) _ l)

dO' g(a2 ; 4 ) 12C 0

(7.48) < e6CoT IVl2 + CiT (e6C^0 T _ l)

• 12Cfi
  since 0 :::; 71 :::; 7 :::; 72 < T. The lemma follows from this and the definition
  (7.44) of C2. D

LEMMA 7.25 (£-geodesic !VP-existence). Let (Mn,g(7)), 7 E [O,T],

be a complete solution to the backward Ricci flow with bounded sectional