240 6. ENTROPY AND NO LOCAL COLLAPSING
(3) Improving on the previous part, show that for any times ti, t2 EI
with ti ::s; t2,μ€ (g (t2), T (t2)) - μ€ (g (ti), T (ti))rt2 r I c:g (t). · 12
2:: lti 2r (t)
1M Rij (t) + V/vjf (t) + 27 (t)J u (t) dμg(t)dt,
where f (t) is the minimizer of We (g (t), · , T (t)) for each t E
[t1,t2].2.4. The behavior ofμ under Cheeger-Gromov convergence. If
(Nn, g) is a complete, noncompact Riemannian manifold, we generalize the
definition of the functional μ to noncompact manifolds byμ (g, r) ~inf { W (g, f, r) : e-f 12 E C': (N), l ( 4wr)-nl^2 e-f dμ = 1},
where W (g, f, r) is defined as in (6.1) and the infimum is taken over smooth
functions with compact support satisfying the constraint.^12LEMMA 6.28 (μ under Cheeger-Gromov convergence). Suppose that we
have (N'f:, 9k, Xk)--+ (N~, g 00 , x 00 ) in the C^00 Cheeger-Gromov sense. Then
for any T > 0,μ (9 00 , r) 2:: limsup μ (gk, r).
k-+oo
PROOF. By definition, there exists an exhaustion {UkhEN of Noo byopen sets with Xoo E Uk and diffeomorphisms k : Uk --+ vk ~ k (Uk) c
Nk with k(xoo) = Xk such that (uk,k [9klvk])--+ (Noo,9oo) in C^00 on
compact sets. Now let e-f/^2 EC~ (N 00 ) with
1
(4wr)-nl^2 e-f dμ 900 = 1.
Noo
Then by the diffeomorphism invariance of W, for all k E N large enough, wehave fNk(4wr)-nl^2 e-f^0 <P"k
1
dμ(w-,;1)* 900 =1 andW ( k [ 9klvk], f, T) = W (9k, f o
Note that although <I>k [ 9klvk] is not defined on the whole N 00 , the entropyW ( k [9klvk], f, T) makes sense since Uk~ supp (e-11^2 ) when k is large.
(^12) Implicitly it is understood in the discussion here that we are considering W as a
function of w ~ (41rT)-n/^4 e-fl^2 E O';;' (N) so that there is no problem with f = oo. As
usual, we use the convention that w^2 log w = 0 when w = O.