1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

3. STEADY AND EXPANDING BREATHER SOLUTIONS REVISITED 211

Note by (5.51) that f satisfies the equation

2b..f -1Vfl^2 + R = >. (g).

On the other hand, R + b..f = 0, so that

1Vfl^2 + R = -).. (g).

However, integrating, we have

->. (g) =JM (1v fl^2 + R) e-f dμ = >. (g),

so that >.(g) = 0 and b..f = 1Vfl^2 = -R. Note that then

0 =JM (b..f - JV JJ

2

) e^1 dμ = -2 JM JV JJ

2

ef dμ

implies that f is constant and hence g is Ricci fiat by (5.57). Alternatively,

we could have argued that since b..f = JV f J^2 ::'.:'. 0, f is subharmonic and

hence constant. D

REMARK 5.29. Even when M is noncompact, we have JV JJ^2 + R is
constant for gradient Ricci solitons; see Proposition 1.15.

3.4. Nonexistence of nontrivial expanding breathers. Recall that

).. (g) is not scale-invariant, e.g., ).. (cg)= c^1 ).. (g). Thus we define the nor-
malized >.-invariant:

(5.58) .\ (g) ~).. (g) ·Vol (M)^2 /n.

It is easy to see that .\ (cg) = .\ (g) for any c > 0, so the invariant .\ is
potentially useful for expanding and shrinking breathers. We shall prove
the monotonicity of .\ (g (t)) under Ricci flow when it is nonpositive. For
this reason it is most useful for expanding breathers.
Recall that by (5.56), we have

(5.59)! >. (g (t)) ::'.:'. 2 JM JRij + ViVjfl^2 e-f dμ,

where -9t>. (g (t)) is defined as the liminf of backward difference quotients.^16

Let V ~ V (t) ~ Volg(t) (M). From (5.59), we compute

!.(g(t)) =! [>.(g(t)) · V(t)^2 !n]

= v2/nd>. + ~v~-1>. dV

dt n dt

::::: 2v^2 1n r 1~j + vivjt1^2 e-1 dμ + ~>.v~-^1 f (-R) dμ,

JM n JM

(^16) This also applies to the time derivatives below in this argument.