1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

228 6. ENTROPY AND NO LOCAL COLLAPSING

Theorem 6.4 then follows from

dW = f (~ - R) vdμ = { (-D* - .6.) vdμ

dt JM at JM

= 2T JM 'Rij + ViVjf - 2 ~9ijl

2

udμ

2:: 0.

Before we .prove (6.22), we need the following lemma concerning the

function u = (47rT)-nl^2 e-f as defined in (6.3) (compare with (5.40)).

LEMMA 6.9 (u is a solution to the adjoint heat equation). The evolution

equation (6.15) off is equivalent to the following evolution equation of u:

(6.23) D*u = 0.

PROOF. We calculate

D

We now show that we can apply the computation in subsection 2.3.2 of

Chapter 5 to derive the evolution equation (6.22) for v.

PROOF OF (6.22). Let /be defined by e-f ~ u. Then!= f+~ log (47rT)

and (g, !) satisfies (5.34) and (5.35). By (5.43), we have that the quantity

V ~ (R+2.6.f-/Vf/

2

) u

= (-2.6.logu - JVlogul^2 + R) u

satisfies the equation

Note that

v = [T ( R + 2.6./ - /V f/

2

) - logu - ~ log(47rT) - n] u

(6.24) =TV - (logu + ~ log(47rT) + n) u.

We compute using D*u = 0 and the general formula

D (ab)= bDa + aD*b - 2 (Va, Vb) - Rab