1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

4. CLASSICAL ENTROPY AND PERELMAN'S ENERGY 217

where we integrated by parts and used Holder's inequality and the Gauss-

Bonnet formula. Thus

(5.69)

d

d (s-2z) ~ _41 (s-2z)2,

s 1rX

wheres~ T~t· From this we conclude that if s 02 Z (so) > 0 for some s 0 < oo,

then s-^2 Z (s)--+ oo ass--+ s1 for some s1 < oo. In other words, if Z (to) > 0

for some to < T, then Z (t) --+ oo as t--+ ti for some ti < T. This contradicts

our assumption that the solution exists on [O, T). Hence Z (t) :::; 0 for all t

and we have proved the following.

THEOREM 5.38 (Hamilton's surface entropy monotonicity). For a solu-

tion of the Ricci fiow on a closed surface with R > 0, we have

dN(t)<O

dt -

for all t E [O, T).

Note that, from (5.69), we have t f--* (T - t)^2 Z (t) is nondecreasing (since

x > 0) and hence there is a constant C > 0 such that (T - t)^2 Z (t) ~ -C

for all t E [O, T). By (5.68),

Z=dN =- { IVRl2 dA+ { (R-r)2dA,

dt JM R JM

and we have

REMARK 5.39. An inequality of the above type is often referred to as a

reverse Poincare inequality.

4.2.2. Entropy formula for Hamilton's surface entropy. Define the po-
tential function f (up to an additive constant) by D.f = r - R. In [97]
the monotonicity of the entropy was proved by relating its time-derivative
to Ricci solitons via an integration by parts using the potential function
(Proposition 5.39 in Volume One). In particular, we have

Note that Rij = !R9ij and r =~JM RdA = (T-t)-^1. We have purposely

written this formula to more resemble Perelman's formulas (5.41) and (6.17).