1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

3. STEADY AND EXPANDING BREATHER SOLUTIONS REVISITED 209

The result now follows from

4

JM \Y'w\~1 dμg1 = 9(91, w) JM R 91 w^2 dμ 91

JM w2dμg1 JM w2dμg1 JM w2dμg1

:::; A. (g1) - minR 91.

D

The monotonicity of :F (g (t), f (t)) under the system (5.34)-(5.35) im-

plies the monotonicity of A. (g (t)) under the Ricci flow.

LEMMA 5.25 (A. monotonicity). If g (t), t E [O, T], is a solution to the

Ricci flow, then

d 2

dt A.(g( t)) 2 ; .x.2 (g(t))'

and A.(g(t)) is nondecreasing int E [O, T]. Here the derivative ft is in the

sense of the lim inf of backward difference quotients.

REMARK 5.26. See the next subsection for the case where A.(g(t)) is not
strictly increasing.

PROOF. Given to E [O, T], let fo be the minimizer of :F (g (to), f), so

that A. (g (to))= :F (g (to), f (to)). Solve

a 2

(5.54) atf = -R-6-f + \\7 f\ , f (to)= fo,

backward in time on [O, to]. Then ft:F (g (t), f (t)) 2 0 for all t:::; to. Since

the constraint JM cf dμ is preserved under (5.54), we have A. (g (t)) :::;
:F (g (t), f (t)) for t :::; to. This, (5.41), and A. (g (to)) = :F (g (to), f (to))
imply both

(5.55) A. (g (t)) :::; :F (g (t), f (t)) :::; :F (g (to), f (to))= A. (g (to))

and the following:

! A (g ( t)) I t=to 2::! :F (g ( t) ' f ( t)) I t=to

(5.56) = 2 JM \Rij + Y'i'Vjf\

2

e-f dμ 9 (to)

2 2 f ~ ( R + 6.f)^2 e-f dμ 9 (to)

}Mn

2 ~ (JM (R + 6-f) e-f dμ 9 (to))

2

= ~A.^2 (g(to)),
n
where f = fo is the minimizer. Hence, from either (5.55) or (5.56), we see
that A. (g ( t)) is nondecreasing under the Ricci flow. D

EXERCISE 5.27. Prove (5.56).