- THE FUNCTIONALSμ AND v 239
Hence we have
(6.58) μ (g (t), T (t)) :::; W (g (t), f (t), T (t))
:::; W (g (to), f (to), r (to))=μ (g (to), r (to))
fort E [O, to].
The above inequality implies(6.59)! lt=to μ (g (t), T (t)) 2::! lt=to W (g (t), f (t), T (t)) z 0
in the sense of the lim inf of backward difference quotients. This inequality
holds for all to E [O, T]. Actually we have! lt=to μ (g (t) 'T (t))
2:: JM 2r (to) IRij (to)+ \li'ljf (to) - 27 ~to)g (to)ijl
2X (47rr (to))-'!,} e-f(to)dμg(to)for the minimizer f (to) of {W(g (to),·, r (to)) : (g (to),·, r (to)) EX}. Hence,
from either (6.58) or (6.59), we have the following monotonicity formula for
μ.
LEMMA 6.26 (μ-invariant monotonicity). Let (g (t), r (t)), t E [O, T], bea solution of (6.14) and (6.16) on a closed manifold Mn with r (t) > 0. For
all 0 :::; ti :::; t2 :::; T, we have
(6.60) μ (g ( t2) , T ( t2)) 2: μ (g (ti) , T (ti)) ·In particular,(6.61) μ (g (t), r^2 ) 2:: μ (g (0), r^2 + t)
for t E [O, T] and r > 0.
The following exercise continues our discussion in subsection 1.3 of this
chapter. Again u ~ (47rr)-'!,J;e-f.
EXERCISE 6.27 (μcinvariant monotonicity). Define the μ€-invariant on
a closed manifold Mn:
μc(g, r) ~inf { Wc(g, f, r): f E C^00 (M), JM udμ = 1}.
(1) Show that for any c > 0,
μ€ (cg, CT) = μ€ (g, T).
(2) Show that if (M, g (t)) is a solution to the Ricci flow on a time in-
terval IC JR, EE JR, and ~; = E with r (t) > 0, then μc (g (t), r (t))
is monotonically nondecreasing on I. That is, for t2 2: ti,
μ€ (g (t2), T (t2)) 2: μc (g (ti), T (ti))·