1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

216 5. ENERGY, MONOTONICITY, AND BREATHERS

REMARK 5.37.

(1) The formula above for ftFm is somewhat reminiscent of Hamilton's

formula for the evolution of the time-derivative dN / dt of his entropy

N (g) ~ J M 2 R log RdA on a positively curved surface evolving by

Ricci flow (see [180]).

(2) We can rewrite (5.66) as

JM (2b.f-IV'fl

2

+ R-;:_) e-f dμ so,

where 'T ~ T - t.

4.2. Hamilton's surface entropy. Recall that the normalized surface

entropy for a closed surface (M^2 , g) with positive curvature is defined by

N(g) ~JM log (RA) Rdμ,

where A is the area. Let (M^2 , g (t)) , t E [O, T), be a solution, on a maximal

time interval of existence, of the Ricci flow on a closed surface with R > 0. In

this subsection we give two proofs of the monotonicity of N(g (t)) ~ N (t).

4.2.1. Hamilton's original proof of surface entropy monotonicity. The

time-derivative of N ( t) is given by

(5.68) dN dt =JM f QRdμ,

where

Q ~ b. log R + R - r

and r is the average scalar curvature. On the other hand, since ~~ = r^2 and

by a similar computation to (Vl-5.38),

f)

ot Q ~ b.Q + 2 (\7 log R, V'Q) + Q^2 + 2rQ.

By the long-time existence theorem (Proposition 5.19 of Volume One), r =

T~t and Area (g (t)) = 47rX (T - t), where x denotes the Euler characteristic

of M. Differentiating (5.68) with respect to time, we compute that Z ~ dd1[

satisfies