- COMPACT FINITE TIME SINGULARITY MODELS ARE SHRINKERS 7
The following is inequality (3) in Lemma 2.1 of [197] (compare with
(6.63) in Part I); again we assume n 2: 3 for simplicity.LEMMA 17.6 (Lower bound for the μ-invariant). If T 2: ~, then(17.25) μ (g, r) 2: (r - i) ,\ (g)-~ log(47rr)-n-nlog Cs (g)+i Rmin (g).
REMARK 17.7.
(1) Note that estimate (17.25) is not true for sufficiently small T. One
reason is because the limit as T-+ O+ of the RHS of (17.25) is equal
to +oo, contradicting (17.47) below.
(2) We see from (17.25) that if ,\ (g) > 0, then limr---+oo μ (g, T) = oo;
see also Lemma 6.30 in Part I.PROOF. We consider the term - JM log (w^2 ) w^2 dμ on the RHS of (17.7).
For any w such that JM w^2 dμ = 1, Jensen's inequality says that if¢: JR-+ JR
is convex and f E L^1 ( w^41 n g) , then
4In particular, taking f = w n-2 and ¢ ( u) = - log u, we have
- JM w^2 log (w^2 ) dμ = -n;
2
JM log ( wn~2) w^2 dμ
2: -n;
2
log (JM w;;-:':2dμ)
= -nlog llwll Ln='l 2n (g)
2: -nlog (Cs (g) llwllw1,2( 9 ))
(17.26) = -n log Cs (g) - ~log ( 1 +JM IV'wl
2
dμ) ,where we used the L^2 Sobolev inequality (17.24).
Since log (1 + x) ::; x for x 2: 0, we have
(17.27)
Combining this with (17.7), we have
n
W (g, f, r) 2: TY (g, w) - 2 log(47rr) - n - nlog Cs (g)- -n1 IV'wl^2 dμ.
2 M