1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

212 5. ENERGY, MONOTONICITY, AND BREATHERS

where f = f (t) is the minimizer of :F (g (t), ·). From this we obtain

t v-^2 /n :t >. (g (t)) 2:: JM IRij + \7i \7jfl^2 e-f dμ

• .!_ { (R + jj,f) e-f dμ. l { Rdμ.
  n}M V )M
  Hence

2v-^1 2 /ndt>.(g(t))2:: d - JM r I Rij+\7i\7jf-;;,(R+jj,_f)gij^1 12 e -fdμ

+ { .!_ (R + ,(j,_f)2 e-f dμ

}Mn

• l JM (R + fj,_f) e-f dμ ·~JM Rdμ.

Recall from (5.53) that

JM (R + fj,f) e-f dμ-:; J ~dμ.

Assuming).. (t) -::;. O, so that JM (R + fj,_J) e-f dμ-::;. 0, we have

1 d _ r I 1

1

2

(5.60)

2

v-^2 /n dt>.(g(t))-JM Rij + \7i\7jf-;;, (R+fj,_f)9ij e-fdμ

2:: l JM (R + jj,_f)

2

e-f dμ - l (JM (R + jj,_f) e-f dμ)

2

2:: 0

since JM e-f dμ = 1. Hence

LEMMA 5.30. Let g (t) be a solution to the Ricci flow on a closed manifold

Mn. If at some time t, >. (t) -:5:_ 0, then

d-

(5.61) dt).. (g (t))

2:: 2v2/n JM 1~j + \7i\7jf - l (R + jj,_f) 9ij 12 e-f dμ 2:: 0,

where V = Volg(t) (M), f (t) is the minimizer for :F (g (t), ·), and the time-

derivative is defined as the lim inf of backward difference quotients. By

(5.61), if -fit>. (g (t)) = 0, then g (t) is a gradient Ricci soliton.

This is reminiscent of the fact that under the normalized Ricci flow,

the minimum scalar curvature is nondecreasing as long as it is nonpositive,

whereas under the unnormalized Ricci flow, the minimum scalar curvature

is always nondecreasing (see Lemma A.20). However these two facts appear

to be quite different in nature.

To apply the above monotonicity to the expanding breather case, we

need to produce a time to where >. (g (to)) < 0. This is accomplished by

looking at the evolution of the volume. Below we also give another proof of

Lemma 5.28 using>. (g (t)).