1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

1. ENERGY, ITS FIRST VARIATION, AND THE GRADIENT FLOW 195

where we think of Re ( e -n!q f (g + hq)) and R ( e -n!q f (g + hq)) as quan-

tities on M since they are independent of the point in Tq. The total scalar

curvatures of (M x Tq, e -n!q f (g + hq)) limit to Perelman's F functional:

Note that

lim { R (e-n!qf (g+hq)) dμ __ (^2 1)

q-+oo J MxTq e n+q (g+hq)

= lim { { R (e-n!qf (g + hq)) e-^2! dμhqdμg

q-+oo JM Jrq

=JM (R+ IV7fl

2

) e-fdμ

= :F(g,J).

gij RiJ = R + flj =Rm - flj + l\7fl^2 ·

There is an analogue of the contracted second Bianchi identity for RiJ and

Rm. In particular we compute

\7iRi} = \7iRij + \7i\7j\7d = t\7jR + '1jtlf + Rjk \7kf

and

1 m 1 2 1 1

2\7jR = '1jtlf-2\7j IV7fl + 2\7jR = 2\7jR+ '1jtlf-'1j\7kf\7kf,

which imply

(5.20)

To understand this formula further, we define

\7m : C^00 (T M ®s T M) ----+ C^00 (T M)

by
(\7*ma)j ~ \7iaij - aji\7d.

LEMMA 5. 7. The operator \7*m is the adjoint of -\7 with respect to the

measure dm = e-f dμ.

PROOF. For any symmetric 2-tensor aij and 1-form bi,

JM aij (-\7i) bje-f dμ =JM bj\7i ( aije-f) dμ

=JM bj (\7iaij - aij'1d) e-f dμ

= r bj (\7*ma). e-f dμ.

JM J

D

Thus (5.20) implies the following, which is the analogue of the contracted

second Bianchi identity.