1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

7. NOTES AND COMMENTARY 121

the effect on the £-length of a path / : [O, f] -+ M by a variation v ( T) of

g ( T). When we compute the variations of associated quantities, we write

Ov. Suppose that v (T) satisfies the backward Lichnerowicz Laplacian heat

equation

0

OT Vij = -.6.LVij·

This implies V = gij Vij satisfies

0

OT V = -.6.V - 2Rc·v.

Fixing the path, considering the variation o g = v, and using

OvR = div (div v) - .6. V - v · Re,

we compute

Ov.C(r) = Ov la'f VT (R(/(T) ,T) + l'Y(T)J~(r)) dT

= la'f VT(div(divv)-.6.V-v·Rc+v(')1,')1))dT

= 1

7

VT (div (divv) + v · Rc-2divv · 'Y + v ('Y,'Y)) dT

• 1

7

VT (:TV+ 2divv · 'Y) dT.

Now

d~(V(r(T),T))= :TV+\7V·')1.

So, from integrating by parts, we have

Ov.C(r) = laT VT(div(divv)+v·Rc-2divv·'Y+v('Y,'Y))dT

• 1

7

VT ( d~ (V (r ( T) , T)) + ( -\7V + 2 div v) · 'Y) dT

(19.74) = 1

7

VT (div (divv) + v · Rc-2 divv · 'Y + v ('Y, 'Y) - ;TV) dT

• Jfv (r (f), f) + 1

7

2VT ( divv - ~\7V) · ')1dT,

where the linear trace Harnack quadratic L ( v, 'Y) defined by (17.91) appears
in the third line. Note that, in the discussion of the Ricci-DeTurck fl.ow in

Volume One, divv - ~\7V =div (v - ~g) is the 1-form on p. 79 in Volume

One; moreover, essentially, the 1-form

Wj ~ 9jk9pq ( r;q -t;q)

defined by (3.34) in Volume One has variation equal to divv - ~\7V.