1547671870-The_Ricci_Flow__Chow

134 5. THE RICCI FLOW ON SURFACES

PROPOSITION 5.39. If (M^2 , g (t)) is a solution of the normalized Ricci

flow on a compact surf ace with R ( ·, 0) > 0, then

dN = - r iv R + R v f i 2 dA - 2 r i Ml2 dA :::; o,

dt }M2 R }M2

where M is the trace-free part of the Hessian off defined in (5.9). In

particular, the entropy is strictly decreasing unless (M^2 , g (t)) is a gradient

Ricci soliton.

PROOF. Recall that b..f = R - r. Expanding the first term on the
right-hand side and integrating by parts, we obtain
(5.26)

r IV' R + RV' fl

2

dA = r (IV' Rl

2

• 2R(R - r) + RIV f 1^2 ) dA.
  }M2 R }M2 R

On the other hand, commuting covariant derivatives and integrating by parts
shows that

{ (R - r)^2 dA = { (b..f)^2 dA

}M2 }M2

hence that

= - f (Y' f, V' b..J) dA

}M2

= - { ( (V' f, b.. V' !) - Re (V' f, V' f)) dA

}M2

= JM 2 ( 1vv fl

2

+~R1v f l

2

) dA,

• 2 /M2 IMl2 dA = !M2 ( (b..f)2 - 2 IV'Y' f l2) dA

= JM2 (~RIV f l

2

• IVY' f l

2

) dA

(5.27) = JM 2 (RIV f l^2 - (R - r)^2 ) dA.

Subtracting (5.26) from (5.27) yields

• 2 r IMl2 dA - r IV' R +RV' f l2 dA = r ((R - r)2 - IV' Rl2) dA,
  }M2 }M2 R }M2 R

whence the result follows by Lemma 5.38. D

COROLLARY 5.40. Let (M^2 , g (t)) be a solution of the normalized Ricci
flow on a closed surface with R (., 0) > 0. Then the entropy is a strictly-
decreasing function of time unless R (-, 0) = r, in which case it is constant
in time.