- SHRINKING BREATHERS ARE SHRINKING GRADIENT RICCI SOLITONS 243
for t E [ti, t2]. Thus f (t) is the minimizer for W (g (t), f (t), T (t)) and
ft W (g (t), f (t), T (t)) = 0, so by (6.17), we have
JM 1~j + \7i\7jf - ;; 12 e-f dμ := 0
for all t E [t1, t2]. We conclude that(6.62) ~j + \7i\7jf - !:] 9" = 0 fort E [t1, t2].
2T
Since a breather is a periodic solution of the Ricci fl.ow (modulo diffeomor-
phisms and homotheties), by the uniqueness of solutions to the Ricci fl.ow
on closed manifolds, the behavior of g (t) on [ti, t2] determines completely
the behavior of g (t) on its whole time interval of existence. This is why,
from (6.62), which is valid on [ti, t2], one can deduce that g (t) is a breather
on [O,T). D
3.2. Asymptotic behavior ofμ and finiteness of v. In the second
proof of Theorem 6.29 given below we need the finiteness of v, which in turn
depends on >..(g(t)) > 0 and the following asymptotic behavior ofμ.
We have shown that for each g and T > 0, μ (g, T) is finite. However we
have yet to study the behavior ofμ (g, T) as T ~ oo or T ~ 0. Recall that
A.(g) = A.1 (-4~ + R) =inf {JM (R + [\7 fl^2 )e-f dμ: JM e-f dμ = 1}.
Sinceμ and Ware modifications of>.. and F, we can prove the following.
LEMMA 6.30 (μ ~ oo as T ~ oo when >.. > 0). If >..(g) > 0, then
lim μ (g, T) = +oo.
T-tOO
REMARK 6.31. The idea of the proof is that when T ~ oo, the F termin the expression (6.4) for W dominates, so if inf F > 0, then inf W ~ oo
as T ~ oo.
PROOF .. By Lemma 6.24, for any T > 0, there exists a C^00 function f 7
with JM(47rT)-nl^2 e-frdμ = 1 such that
μ(g, T) = W(g, JT, T) =JM [T (R + [\7 f 712 ) + f 7 - n] (47rT)-nl^2 e-frdμ.
We add a constant to f 7 so that it satisfies the constraint for F (g, ·) (instead
of W(g, ·, T)) and we define f ~ f 7 +~ log (47rT) so that JM e-1dμ=1. Then
by the logarithmic Sobolev inequality (e.g., Corollary 6.38 with b = 1), we
have
μ (g, T) = JM [ T ( R + I \7 f I^2 ) + f - ~ log ( 41fT) - n J e-l dμ
2: JM (TR+ (T -1) l\7fl^2 ) e-1 dμ-~log (47rT) - n - Ci(g)
2: ( T - 1) JM ( R + I \7 f I^2 ) e-l dμ + Rmin - ~ log ( 41fT) - n - C 1 (g).