1547671870-The_Ricci_Flow__Chow

8. SURFACE ENTROPY 135

PROOF. If dN/dt = 0 at some time to E [O, oo), then M(·,to) = 0.
By equation (5.7), g (to) is a gradient Ricci soliton. Hence R (·,to) = r is
constant by Proposition 5.21, and thus g (t) = g (to). D

8.2. The case that R (-, 0) changes sign. We want to extend our
previous entropy estimates to initial metrics on 52 or !RlP'^2 whose curvature
changes sign. We first need to define a suitable modification of the notion of
entropy. By passing to the double cover if necessary, we may assume M^2 is
diffeomorphic to 52. Recall that the ODE corresponding to the PDE satisfied
by R is
d
dt s = s( s - r).

Recall that r > 0 on M^2 ~ 52 , and let s(t) be the ODE solution with initial

condition s(O) =so < Rmin(O) < 0, namely

r

s(t) =.

1 - ( 1 - ; 0 ) ert

(5.28) (so < Rmin(O) < 0)

Then the difference of R and s evolves by

a

at (R- s) = 6. (R - s) + (R - r + s) (R - s).

Since Rmin (0) - so > 0, the maximum principle implies that R-s > 0 for as

long as the solution of the normalized Ricci fl.ow exists. This suggests that

we define a modified entropy by

(5.29) N (g (t) 's (t)) ~ r (R - s) log (R - s) dA.

}M2

LEMMA 5.41. Let (M^2 , g (t)) be a solution of the normalized Ricci flow

on a closed surface with an arbitrary initial metric satisfying r > 0. Then

~N= f (-\7R\

2

+(R-s)(R- r+s+slog(R- s))) dA.

dt }M2 R - s

PROOF. Put P ~ R - s. Noting that

!!_(PdA) = (6.P+(R-r+s)P) dA+P(r-R) dA= (6.P+sP) dA,

at

we integrate by parts to obtain

• d NA = J [(6.P+(R- r+s)P)+(logP)(6.P+sP)] dA
  dt M2

= JM

2

[-~ \7 P\^2 + (R - r + s +slog P) P ] dA.

D