1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

1. THE ENTROPY FUNCTIONAL W AND ITS MONOTONICITY 225

i.e., the variation preserves the measure (47rT)-nl^2 e-f dμ on M, we obtain

from ( 6. 9) the gradient flow (assuming ~; = -1)

a

(6.10) at9ij = -2 (Rij + \7/\Jjf),

af n

(6.11) at = -f:lf - R + 2T,

(6.12) ~: = -1.

LEMMA 6.3 (Entropy monotonicity for gradient flow). If (g(t), f(t), T(t))

is a solution to the system (6.10)-(6.12), then

d

dt W(g(t), f (t), T(t))

(6.13)

PROOF. This follows directly from substituting

Vij = -2 (Rij + \li\ljf)'

n

h = -f:lf-R+ 2T'

( = -1

into (6.9) and using the facts

~ -h-;c = O and JM (!:lf-l\lfl

2

) e-^1 dμ = o.

D

1.2. Coupled evolution equations associated to W and mono-

tonicity of W.

1.2.1. The coupled evolution equations associated to W. As in Chapter

5, there is a system of evolution equations for the triple (g, f, T) (see (1.3)

of [297])

(6.14)

(6.15)

(6.16)

a

at9ij = -2Rij,

af · n

at = -f:lf + 1v11

2

• R+ 27 ,

dT = -l
dt '
whose solution differs from the solution to (6.10)-(6.12) by diffeomorphisms.
This leads to the following theorem, which says that
d
dt W(g(t), f(t), T(t)) 2: 0.

Another motivation for studying this system of equations, from considering
gradient Ricci solitons, was discussed in Section 8 of Chapter 1.