- SOME FURTHER CALCULATIONS RELATED TO :F AND W 275
LEMMA 6.87. Under the equations gtgij = -2Rij and
{)
8t Xi = jj.dXi - Y'iR + 2 (\7 xX)i,we have( :t + jj. - 2X · \7) ( R + 2gij\7iXj - JXJ
2
)I1 12= 2 Rij + 2 (Y'iXj + Y'jXi) -^3
2
JY'iXj - Y'jXiJ^2 •
This extends the calculation (6.22) of Perelman.
7.1.2. An extension of the monotonicity formula. A generalization of
Perelman's energy monotonicity formula is given by the following.LEMMA 6.88. If gtgij = -2Rij and
~~ = jj.f +a ( R + 2/j.f - JV' fJ
2
),
for some a E JR, then( ~ - (2a + 1) fj. + 2a\7 f · \7) ( R + 2/j.f - JV' fJ^2 ) = 2 J~j + \7iY'jfJ^2.
and
PROOF. We have by direct calculation,:t JV'fJ^2 = 2RijY'dY'jf +2\i'f · \7 [/j.f +a (R+ 2/j.f-JV'fJ
2)]
= (2a + 1) jj. JV'fJ^2 - 2(2a+1) JY'iY'jfJ^2 -4a~jY'dY'jf
- 2\i'f · \7 [a ( R-JV'JJ
2) J
:t (jj.J) = 2~jY'iY'jf + jj. ( jj.j +a ( R + 2/j.f - JV' JJ^2 ))
= (2a + 1) fj. (Dl.J) + 2RijY'iY'jf + fj. (a ( R-J\7 JJ^2 )).
Hence
( :t - (2a + 1) jj.) ( R + 2/j.f - JV' JJ
2
)= -2a/j.R + 2 JRijJ^2
- 4RijY'iY'jf + 2/j. (a ( R - JV' JJ
2
))- 2(2a+1) JY'iY'jfJ^2 + 4aRijY'dY'jf-2\i'f · \7 [a ( R-JY'JJ
2) J
= 2 JRijJ^2 + 4Rij Y'iY'jf + 2 JY'iY'jfJ^2 - 2a\7 f · \7 ( R + 2/j.f - JV' JJ^2 )
and the lemma follows. 0
From this we deduce the following.