1547671870-The_Ricci_Flow__Chow

136 5. THE RICCI FLOW ON SURFACES

When the initial metric has strictly positive curvature, Corollary 5. 40
shows that the entropy is decreasing. Not surprisingly, one can no longer
prove this result for initial metrics whose curvature changes sign. However,
one can show that the entropy is uniformly bounded from above, which is
what we need for our applications.

PROPOSITION 5.42. Let (M^2 , g (t)) be a solution of the normalized Ricci

flow corresponding to an arbitrary initial metric with r > 0. Then the
modified entropy evolves by

.!!:__ N = - r iv R + ( R - s) v Ji 2 dA - 2 r i Ml2 dA

dt } M2 R - s } M2

(5.30) - s JM 2 (IV fl^2 + s - r - (R - s) log (R - s)) dA,

where M is the trace-free part of VV f.

PROOF. Expand the first term on the right-hand side and integrate by
parts to get

r 1 v R + ( R - s) v f 12 dA

}M2 R-s

= { ( IVRl

2

• 2R(R- r)+(R-s)IVfl^2 ) dA.
  }M2 R-s

But as we observed in (5.27),

• 2 JM 2 1Ml

2

dA= JM 2 (R1Vfl

2

• R(R-r)) dA.

Subtracting these identities yields

• 2 r IMl2 dA - r IV R + (R - s) v fl2 dA
  }M2 }M2 R

= L (R(R-r)+ s l'V/1

2

• ~~I:) dA

whence the proposition follows by Lemma 5.38. 0

Since s < 0, we expect the last line of equation (5.30) to be positive. The

most difficult term to control is f M2 IV f l^2 dA. To obtain an upper bound

for the modified entropy, therefore, we shall estimate its integral with respect

to time.

LEMMA 5.43. There exists a constant C depending only on g (0) such

that if the solution (M^2 , g ( t)) of the normalized Ricci flow with r > 0 exists

for 0 :S t < T, then

{Te-rt r IV f l^2 dA dt :S C.

Jo }M2