1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

7. SOME FURTHER CALCULATIONS RELATED TO :F AND W 281

and let v be a symmetric 2-tensor. If g (s) = g +sh, then

d

2

-d 2 I v (g (s)) = ( T A)^1 A ( --^1 2. 2^1 2)

s s=O Vol M M^2 l\7vl + ld1vvl - -^2 l\7wl dμ

• Vol ( M) L ( Il;;.,v;;vt.+ 4~ w') dμ

• ~ (

1

f Vdμ)

2

2n Vol (.M) JM '

where w is defined uniquely by

~w +:!!._~div (divv),
2T JM wdμ= 0.
7.3. A matrix Harnack calculation for the adjoint heat equa-
tion. We also have the following Harnack-type calculation (see [287]). We
may think of equation (6.113) below as a Bochner-type formula which says
that the evolution of the modified Ricci tensor is given by a backward Lich-
nerowicz Laplacian heat operator with Hamilton's matrix Harnack quadratic
as the main term on the RHS.

PROPOSITION 6.95 (Matrix Harnack formula for adjoint heat equation).

Under the system

we have

(6.113)

{)

ot9ij = -2Rij,

Of 2 nE

ot = -~f-R+ l\7fl - 2T'

dT

dt = E,

( :t + ~L - 2\7 f · \7) ( \7i\7jf + Rij + 2 :9ij)

= 2 ( ~LRij - ~\7i\7jR + RikRjk + ;T Rij)

• 2 ((Pi£j + Pjei) \7d + Rkije\7d\7kf)

• ( \7i\7kf +Rik+ 2ET9ik) (-\7j\7kf + Rjk + 2:9jk)


• (-\7i\7kf +Rik+ 2 :9ik) (\7j\7kf + Rjk + 2 :9jk) ·

PROOF. We have

(%t +~L) \7i\7j¢=\7i\7j (~~ +~¢)-2 (:trfj) \7e¢