To deal with the expanding case consider a scale invariant version
¯
λ(gij ) = λ(gij )V 2/n (gij ). The nontrivial expanding breathers will be ruled
out once we prove the following
Claim ¯
λ is nondecreasing along the Ricci flow whenever it is nonpositive;
moreover, the monotonicity is strict unless we are on a gradient soliton.
(Indeed, on an expanding breather we would necessarily have dV /dt > 0
for some t
∈[t1 , t2 ]. On the other hand, for every t, −
d
dt logV =
1
V
􏰂
RdV
≥
λ(t), so ¯
λ can not be nonnegative everywhere on [t1 , t2 ], and the claim ap-
plies.)
Proof of the claim.
d¯
λ(t)/dt
≥ 2V
2/n
􏰂 |
Rij +
∇i ∇j f |
2
e−f dV + 2
n V (2−n)/n λ
􏰂 −
RdV
≥
2V 2/n [
􏰂 |
Rij +
∇i ∇j f −
1
n (R +
△f )gij |
2
e−f dV +
1
n (
􏰂
(R +
△f )
2
e−f dV
− (􏰂 (R + △f )e−
f
dV )2 )]
≥ 0,
where f is the minimizer for
F .