1547671870-The_Ricci_Flow__Chow

8. SURFACE ENTROPY 133


9. Surface entropy

Since we have not been able to employ the maximum principle to obtain
a uniform upper bound for the curvature in the case that r > 0, we shall
consider integral quantities. Perhaps the most important such quantity for
the Ricci flow on surfaces is the surface entropy.

8.1. The case that R (·, 0) > O. We first consider the case that R > 0.
The surface entropy N is defined for a metric of strictly positive curvature
on a closed manifold Mn by

N(g) ~ f RlogRdμ.

}Mn

The reason this quantity is called an entropy is because it formally resembles
certain classical entropies, each of which is the integral of a positive function
times its logarithm. Our sign convention is opposite the usual one, and so we
shall show that the entropy is decreasing (instead of increasing) under the
normalized Ricci flow. We first compute the time derivative of the entropy.

LEMMA 5.37. If (M^2 ,g(t)) is a solution of the normalized Ricci flow

on a compact surface, then

(5.24) gt (RdA) = t:,.RdA.

PROOF. By (5.2) and (5.3), we have

gt ( R dA) = (gt R) dA + R (gt dA)

= [!:::,. R + R (R-r)] dA + R (r - R) dA.

D

LEMMA 5.38. If (M^2 , g (t)) is a solution of the normalized Ricci flow

on a compact surf ace such that R ( ·, 0) > 0, then the entropy evolves by

(5.25) dN = - f \7 R\

2

dA + f (R - r)^2 dA

dt }M2 R }M2

PROOF. Using (5.24) and integrating by parts, we calculate

dN = f ( ~ log R) R dA + f log R · ~ ( R dA)

dt }M2 ut }M2 ut

= f [!:::,. R + R(R - r)] dA

}M2

+ f log R · t:,. R dA

}M2

= f R(R-r)dA- f \7R\

2

dA.

}M2 }M2 R

D