- STEADY AND EXPANDING BREATHER SOLUTIONS REVISITED 207
(v) (Scaling)
).. (cg) = c -l).. (g).3.2. The monotonicity of>.. Let (Mn, g (t)), t E [O, T], be a solu-
tion of the Ricci fl.ow on a closed manifold. In this subsection we discuss
some properties related to the continuity and monotonicity of >.(g(t)). Such
properties are key to the proof of the nonexistence of nontrivial expanding
or steady breathers. First we show that >.(g(t)) is a continuous function on
[t 1 , t 2 ]. This is a consequence of the following elementary result (see also
Craioveanu, Puta, and Rassias [118] or Chapter XII of Reed and Simon
[310]).^10
LEMMA 5.24 (Effective estimate for continuous dependence of ).. on g).
If gl and g2 are two metrics on M which satisfy
1
1
+ Eg1 :S g2 :S (1 + c) gl and R (g1) - E :SR (g2) :SR (g1) + E,
then^11
).. (g2) - ).. (g1)::::; ((1 + c)~+l - (1 + c)-n!^2 ) (1 + cr/^2 (>. (g1) - minR 91 )
+ ((1+5) max IRgz - Rg1 I+ 25max IRg11) (1 + cr/^2 '
where 5 -+ 0 as E -+ 0.^12 In particular, ).. : 9J1:et-+ JR is a continuous function
with respect to the C^2 -topology.
PROOF. The proof is straightforward but slightly tedious. First note
that (1 + E )-n/^2 dμ 91 '.S dμ 92 '.S (1 + E r/^2 dμ 91. If W is a positive function
on M, then in view of (5.46), we compute (writing a· b-c · d =a (b - d) +
(a-c)d)
JM w^2 dμ 91 Q(g2, w) - JM w^2 dμ 92 Q(g1, w)= 4 JM w
2dμ 91 (JM IVwl~ 2 dμ 92 - JM IVwl~ 1 dμ^91 )
- 4 (JM w
2
dμ 91 - JM w2
dμ^92 ) JM IVwl~ 1 dμ^91- JM w
2dμ 91 (JM R 92 w
2
dμ 92 - JM R 91 w2
dμ^91 )- (JM w2dμg1 - JM w2dμg2) JM Rg1 w2dμgu
lOThanks to [231] for this last reference.
llTo denote the dependence on gi, we use the subscript Yi instead of (gi). So R 91 =
R (g1).
12see the proof for an explicit dependence of Jone:.