- STEADY AND EXPANDING BREATHER SOLUTIONS REVISITED 213
LEMMA 5.31. Expanding or steady breathers on closed manifolds are
Einstein.PROOF. Let (Mn, g (t)) be an expanding or steady breather with g(t 2 ) =acp* g( ti) for some ti < t2 and a ;:::: 1. We have .\ (g ( t2)) = ,\ (g (ti)). Let
V (t) ~ Volg(t) (M). Since V (t2) ;:::: V (ti), we have for some to E (ti, t2),
0::::; dd I logV (t) = _J-;: ~~μ(to)::::;-,\ (g (to)).
t t=to toBy Lemma 5.30, if g (t) is not a gradient Ricci soliton, then ft.\ (g (to)) >
0 and we have.\ (g (t~)) < 0 for some t~ <to. Now since.\ (g (t)) is increasing
whenever it is negative, we have
.(g(t2)) = .(g(t1))::::;), (g (t~)) < 0,
which implies A (g (t)) ::::; A (g (t2)) < 0 for all t E [t1, t2] · Hence .\ (g (t)) is
nondecreasing, which implies.\ (g (t)) is constant. By (5.61), we have
1
~j + ViVjf - - (R + b.f) gij = 0,
n
and since we are in the equality case of (5.60), we also have(5.62) R + b.f = C1 (t) =canst (depending on time).
That is, we still conclude that g (t) is an expanding or steady gradient Ricci
soliton.
Now let (Mn, g ( t)) be an expanding or steady gradient Ricci soliton.
Recall
2b.f + R-IV fl^2 = C2 (t) =canst.
This, combined with (5.62), implies
b.f - IV fl^2 =canst.
Since
JM (b.f - IV !1^2 ) e-f dμ = 0,
we have b.f -1Vfl^2 = 0. Thus, by the strong maximum principle (or since
now 0 = JM ( b.f - IV f1^2 ) ef dμ = -2 JM JV fl^2 ef dμ), we conclude that
f = canst. Hence ~j - ~ Rgij = 0 and gij is Einstein. (When n = 2, our
conclusion is vacuous.) DREMARK 5.32. As a corollary of the above result, we again see that
expanding or steady solitons on closed manifolds are Einstein. In the case
of shrinking solitons on closed manifolds, using the entropy functional, we
shall see in the next chapter that they are necessarily gradient shrinking
solitons.