12 17. ENTROPY, μ-INVARIANT, AND FINITE TIME SINGULARITIES
PROOF. STEP 1. v (gi) is bounded above by a negative constant. By(17.42) below there exists f > 0 such that
μ (g 00 , 7) < 0.
On the other hand, by Lemma 17.15 below,
_lim μ (% f) =μ(goo, f),
i-700
so that
1
V (gi) ::=:; μ (gi, f) ::=:; 2,μ (goo, f) < 0
for i sufficiently large. Thus
(17.34)for all i EN U {oo} and some Eo > 0.
STEP 2. Properties ofμ(% T).
(i) By Lemma 17.15 again, we have for any C > 1,
(17.35) μ (gi, T) ---t μ (g 00 , T)uniformly with respect to TE [ c-^1 , CJ.
(ii) Since ,\ (gi) ---+ ,\ (g 00 ) > 0 and Cs (gi) and Rmin (gi) are uniformly
bounded, Lemma 17.6 implies that there exists C1 E rn, oo) such that
(17.36) μ (9i, T) 2:: 0for all T 2:: C1 and i E NU {oo}.
(iii) By the proof of Proposition 17.20 below, we have that for any se-
quence Ti ---+ 0 there exists a subsequence such that
_lim μ (9i, Ti) = 0.
i-700This implies that for any E > 0 there exists T ( E) > 0 such that
(17.37) μ (gi, T) 2:: -E
for all i EN and TE (0, T (E)].
STEP 3. Completion of the proof. Equation (17.33) now follows from
combining (17.34), (17.35), (17.36), and (17.37). D
Finally, we give the proof ofLEMMA l'l.15 (Continuous dependence ofμ (g, T) on g). For any n 2:: 2,
C < oo, and E > 0, there exists 5 > 0 such that if Mn is a closed manifold
and if g and g are Riemannian metrics such that
then
(1) Ravg (g) ::=:; C,
(2) Vol(§) :::::; C,
(3) IRg - R_gl :::::; 5,
(4) Jg-§19:::::;5,forT E [c-^1 ,c].