1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

1. THE ENTROPY FUNCTIONAL W AND ITS MONOTONICITY 229

that

since

D* [ (logu + ~ log(47rT) + n) u]

= uD* log u +log uD*u - 2 ('V log u, 'Vu) - Ru log u + .!!_u

2T

=-Ru - f'Vloguf^2 u + .!!_u,

2T

D* logu = t (-:t -~) u + f'Vloguf^2 + Rlogu

= -R + f'Vloguf^2 + Rlogu.

From (6.24), we compute

D*v = TO*V + V + f'Vloguf^2 u +Ru - .!!_u

2T

= -2TfRij - \li'Vj loguf^2 u

• (-2~logu - f'Vloguf^2 + R) u

+ f'V log uf 2 u +Ru - .!!_u

2T

= -2TfRij - 'Vi 'Vj · log uf^2 u + 2 (-~log u + R) u - .!!_u. 2T

Completing the square, we obtain (6.22). D

1.3. A unified treatment of energy F and entropy W. We finish

this section with several exercises. In total, this unifies part of the dis-
cussion of shrinking, steady, and expanding gradient Ricci solitons, which
correspond to entropy, energy, and expander entropy on a closed manifold
JVt.n (see the definition below), respectively.^7 In this subsection we use the
following convention: c: E IR, and if c: =/= 0, we take T (t) = c:t, whereas if
c: = 0, we take T (t) = 1. We only consider all t E IR such that T (t) > 0.
Let

(6.25)

Define the c:-entropy by

(6.26) Ws(g,f,T) ~JM (T (R+ f'Vff^2 )-c:(f-n)) (4KT)-nl^2 e-fdμ

=JM Veudμ,

(^7) Notation: For the most part we use a hat on M to emphasize that we are considering
a fixed metric instead of a solution to the Ricci flow.