1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

1. THE ENTROPY FUNCTIONAL W AND ITS MONOTONICITY 227

to (5.31), so that we get

d:F r 2 -! n

dt = 2 JM IRij + Y'iY'jfl e dμ - 27 F.

Similarly, (5.64) becomes

dN = -:F + !!_ { e-f dμ - !!_N

dt 2T JM 2T

under (6.10)-(6.11). Hence, by (6.5), i.e.,

W =(47rT)-nf^2 (T:F(g, f) +N) - n JM udμ

and JM udμ = const, we have

d: =! [(47rT)-n/^2 (T:F +N)]

= (47rT)-nf2 (!!_ (T:F +N) -:F + 7 d:F + dN)

2T dt dt '

so that

(47rTr1^2 d; = 2T JM IRij + \7i\7jfl^2 e-f dμ - 2:F +~JM e-f dμ

= 2T JM 'Rij + \7i\7jf - 2 ~9ijl

2

e-f dμ.

1.2.2. Second proof of the monotonicity of W from a pointwise estimate.
Analogously to subsection 2.3.2 of Chapter 5 we again derive (6.17) using a
pointwise evolution formula. Let^6

(6.20)

In Part II of this volume we shall see that v is nonpositive when u is a
fundamental solution (for this reason vis also called Perelman's Harnack
quantity). Note that

(6.21) W(g,f,T)= JMvdμ.

We shall show the following below.

LEMMA 6.8 (Perelman's Harnack quantity satisfies adjoint heat-type
equation). Under (6.14)-(6.16)

(6.22) D*v = -2T IRij + \7i\7jf - ;

7

9ij,

2

u,

where D* = -gt -.6. + R is the adjoint heat operator defined in (5.39).

(^6) This quantity vis not to be confused with v = og.