264 9. TYPE I SINGULARITIES
If v < - A, then lvl :::; R/3 by Theorem 9.4; so 1 :::; N:::; 2, and we have
IRml :::; lvl + lμI + l>-1 :::; v + μ + >. + 2N lvl :::; ( 1 + ~N) R.
DLEMMA 9.11. The function F satisfies a differential inequality of the
form
a 2(1- c)
at F :::; b..F + R + P (\! F, \! R) + H,where
(T - t)c: [ o 2 ]
H ~ 3 _c: (B1 + B2 - G1) IRml IRml - G2.(R+ p)
Here, B1 ~ C1p and B2 ~ C2c IRml are bad terms, while G1 ~ r°:-t and
G2 ~ 2P are good terms. The function P is defined in ( 6. 39).PROOF. Recall that
0
IRml^2 = 3 1 [ (>. - μ)^2 + (>. - v)^2 + (μ - v) 2]and setL; ~ >,3 + μ3 + v 3 + 3>.μv - (>.2μ + A2v + μ2 A+ μ2v + v2 A + v2μ)
= 2 ~R IR~l^2 - A(μ - v)^2 - μ (>. - v)^2 - v (>. - μ)^2.0
Taking <.p = (T-t)c: I Rml^2 , '1jJ = R + p, a = 1, and f3 = 2 - E in Lemma
6.33, we get0
(T tr IRml^2
-(2-E) - 3 _ (>.^2 +μ^2 +v^2 +>.μ+>.v+μv)
(R + p) c:
0
_ ( 2 _ c) ( 3 _ c) (T - t)c: 1Rml2
IV R l2(R + p)4-c:
+ 2 (2 - c) (T-t~~c: / VIR~l^2 , \! R).
(R + p) \