322 7. THE REDUCED DISTANCEAs before, if L (·, T) is not C^2 at q, by our convention the above inequality
is understood in the barrier sense.6. Equations and inequalities satisfied by L and £
In this section (Mn,g (T)), T E [O, T], will be a complete solution to
the backward Ricci flow satisfying the curvature bound max {I Rm I , I Rel} :SCo < oo on M x [O, T] , and p E M will be a basepoint. Given a point
q EM and 7 E (0, T), let 'Y: [O, 7]---+ M be a minimal £-geodesic from p to
q and let X ( T) ~ ~:;'..6.1. The ..C-length integrand and the trace Harnack quadratic.
Now using the £-geodesic equation (7.32), we compute the evolution of the
..C-length integrand asd~ (R ('Y (T) 'T) +IX (T)l~(T))
&.R
= OT + \l.R · X + 2Rc (X,X) + 2 (\lxX,X)= ~: + \l.R. x + 2Rc (X,X) + ( \l.R-4Rc (X) - ~x,x)
&.R 1 2
= ~ + 2\1.R · X - 2 Re (X, X) - - IXI.
UT T
Hence(7.77) d~ ( R + 1x12) = -H (X) - ~ ( R + 1x1
2
),where H (X) is defined in (7.73). In another form, we haveRecall thatK = K ("!, 7) =for T^312 H (X) dT
defined in ('7.75). Multiplying (7.77) by T^3 /^2 and integrating (by parts for
the second equality), we getHence
-K('Y,7)
=for [T3/2 d~ ( .R + 1x12) + Ti;2 ( .R + 1x12) J dT
= 73 /^2 ( .R ('Y (7), 7) +IX (7)1^2 ) - ~for T^1!^2 ( .R + 1x1^2 ) dT
= 73 /^2 ( .R ('Y (7) '7) +IX (7)1^2 ) - ~..c ('Y).