1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

4. CLASSICAL ENTROPY AND PERELMAN'S ENERGY 215

that is,

T < n { dm.

• 2.rm(g(O)) JM
  The proposition is a consequence of the following.

LEMMA 5.35 (Monotonicity formula for the classical entropy N). If

(g (t), f (t)), t E [O, T), is a solution of the gradient flow (5.29)-(5.30) on a

closed manifold Mn, then

dt_rm d (g(t)) ~:;;, 2 ( JM { dm )-1 _rm (g(t))^2 ,

(5.66) ?(g(t)) ~ 2 (Tn-t) JM e-f dμ.

By (5.64), this implies the following entropy monotonicity formula:

(5.67)! (N-(~JMe-fdμ)log(T-t)) ~O.

REMARK 5.36. Following §6.5 of [356], we may adjust the entropy quan-

tity on the LHS of (5.67) by adding a constant and define

N ~ N-(~JM e-f dμ) (log [47r (T - t)] + 1).

Then we still have a{[ ~ 0, whereas N has the property that for a funda-

mental solution u = e-f limiting to a 5-function as t----+ T, we have N----+ 0

as t ----+ T.

PROOF OF THE LEMMA. From (5.28), we have

d

d? (g(t)) = 2 { [Rij + ViVjf[^2 dm ~ ~ { (R + b.f)^2 dm

t JM n JM

~ HL (R+t>f)dm )'/ L dm

= ~ (JM dm )-l _rm (g(t))^2.

The solution of the ODE

dx 2

-=ex

dt

with limt--+T x (t) = oo is

1

x (t) = c (T - t)'

Hence, taking c =~(JM dmr

1

, we get

_rm (g(t)) ~ 2 (Tn-t) JM dm.

D