192 6. THREE-MANIFOLDS OF POSITIVE RICCI CURVATURE
where P may be written in terms of the eigenvalues >.., μ, v of Rm as
(6.39) p ~ ,>..2 (μ _ v)2 + μ2 (>.. _ v)2 + v2 (>.. _ μ)2 ~ O.By the maximum principle, we only need to show that Q :::; 0 to establish the
0
lemma. Notice that the good term is -P, while the bad term IRml^2 IRml^2 is
scaled by E. So to prove Q :::; 0, one first observes that p ~ o^2 1Rml^2 IR~l^2
if Re ~ oR · g for some constant o > 0. Then one uses Corollary 6.28 to
conclude that the inequality Re ~ oR-g is preserved if it holds at t = 0. Thus
we can take c = '5^2 for a sufficiently small positive constant o = o (g 0 ). 0
REMARK 6.32. The hypothesis of strictly positive Ricci curvature isnecessary for either proof to work, since Q > 0 if A > 0 and μ = v = 0 at a
point. This failure is related to the fact that 51 x 52 accepts no metric of
positive Ricci curvature. (Indeed, by Myers' Theorem, a complete manifold
Mn admits such a metric only if its universal cover is compact.)
Equation (6.38) is derived in the following two lemmas:LEMMA 6.33. If cp is a nonnegative function and 'ljJ is a positive function
on space and time, then(f.) ) ('PO'. ) cpa.- 1 ( f.) ) 'PO'. ( f.) )
fJt - Li 'lj;/3 = a----;p3 fJt - Li cp - (3 ¢/3+1 fJt - Li 'ljJ
cpa.-2 'PO'.- a (a - 1) ----;p3 l'Vcpl^2 - /3 ((3 + 1) ¢/3+ 2 l'V'l/Jl^2
cpa.- 1
+ 2af3 ¢/3+l ('Vcp, \7¢).PROOF. Straightforward calculation. 0
0
LEMMA 6.34. The function f ~ R.s-^2 1Rml^2 satisfies the differential in-
equality~~ :::; 6.f +
2
(l;; E) (\7 f, \7 R) + 2Q.
PROOF. Recall that in the time-dependent orthonormal moving frame
{ ei}, we have
f.)
fJt Rm= 6.Rm+Rm^2 +Rm#.
Hence
:t 1Rml^2 = 2 ( 6. Rm+ Rm^2 + Rm#, Rm)
= 6. IRml^2 - 2 l\7Rml^2 + 2 (Rm^2 +Rm #,Rm)
= 6. IRml^2 - 2 l\7 Rml^2 + 2 (>..^3 + μ^3 + v^3 + 3>..μv).