- IMPROVED VERSION OF NLC AND DIAMETER CONTROL 271
LEMMA 6.80. Let u be a positive solution to the heat equation(:t-.6.)u=O
on a fixed Riemannian manifold (Mn,9). If f = -logu, that is, u = e-f,
then(6.102)PROOF. (1) Using %d = .6.f - l\7 fl^2 , we calculate
! JM l\7fl
2
udμ =!JM (.6.J)udμ=JM ( 2.6.f - l\7 fl
2
) .6.udμ= 2 JM (.6.f .6.u + \7i\7jf\7if\7ju) dμ.
Now integrating by parts yieldsJM .6.f .6.udμ = - JM \7 f · \7.6.udμ
= - JM (\7 f · .6. \7u - Rij\7if\7ju) dμ
= JM (\7\7 f · \7\7 u - uRij \7 if\7 j f) dμ
=JM (-u l\7i\7jfl^2 - \7i\7jf\7if\7ju - uRij\7if\7jf) dμ.
The lemma immediately follows from combining the above two formulas.
(2) Alternatively, one can integrate the following formula to get (6.102)
(see (2.1) and Lemma 2.1 in [283]):(:t -.6.) (u(2.6.f-l\7fl
2
)) =-2ul\7i\7jfl2-2uRij\7d\7jf,
which follows directly from the calculation:
:t ( 2.6.f - l\7 fl
2
) = 2.6. ( .6.f - l\7f12) - 2\7 f. \7 ( .6.f - l\7f1
2
)= .6. (2.6.f-l\7fl^2 ) - 2\7f. \7 (2.6.f-l\7fl^2 )
- .6. l\7f1^2 + 2\7 f. \7.6.f
= .6. (2.6.f-l\7fl^2 ) - 2\7f. \7 (2.6.f-l\7fl
2
)- 2 l\7i\7jfl^2 - 2~j\7d\7jf.
D
Now we can prove the following weaker version of Cheng's Theorem 6.79.