- LOCAL PINCHING ESTIMATES 193
Since the scalar curvature is R =tr (Rm) =tr (Re) = >.. + μ + v, it follows
from (6.33) that
(6.40)a
at R = b..R + 2 IRcl2
= b..R + >..^2 + μ^2 + v^2 + >..μ + >..v + μv,
hence that:t (R2) = b.. (R2) - 2 l\7 Rl2
+ 2 (>.. + μ + v) (>..^2 + μ^2 + v^2 + >..μ + >..v + μv).
0 2
Now since JRmJ^2 = IRml - R^2 /3, we compute thata
a IR~l^2 = b.. JRmJ^2 - ~b.. (R^2 ) - 2 IV RmJ^2 + ~JV RJ^2
t 3 3
+ 2 (>..3 + μ3 + v3 + 3>..μv)- ~ (>.. + μ + v) (>..2 + μ2 + v2 + >..μ + >..v + μv)
0 0
= b..JRmJ^2 - 2JVRmJ^2
4
- 3 (>..3 + μ3 + v3 + 3>..μv)
4
_ - (>..2μ+>..2v+μ2>..+μ2v+v2>.. +v2μ).
3
0
JRmJ^2- (2 - c) R3-E: (>..2 + μ2 + v2 + >..μ + >..v + μv)
0 - (2 - c:) (3 - c:) 1Rml2 J\7 Rl2 + 2 (2 - c) / VJR~J2 \7 R).
R4-E: R3-E: \ ,
Simplifying and combining terms, we obtain
~ f = b..f + 2 (1 - c) j \7 ( JR~J2) \7 R) + 2c JR~l2 JRcJ2
at R \ R^2 - E: ' R^3 - E:
2 \ (
0
)0
\2
JR0
12
1v Rl2
- R 4 -E: R \7Rm - \7RQ9Rm - c:(l-c:) mR 4 -E:
_ R^4
3 -E: [>..2μ2 + >.2v2 + μ2v2 (>..2μv + μ2>..v + v2>..μ)].