1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

7. SOME FURTHER CALCULATIONS RELATED TO :F AND W 283

EXERCISE 6.98. Under the system of equations

8

at9ij = -2Rij,

8

at X = -LlX + Re ( X) - \7 R + 2 (\7 X* X) ,

dT

dt = E,

generalize the above calculation to find

(%t + LlL -X · \7 -Cx) (2Rij + Y'iXj + Y'jXi + ~9ij).

There is also a corresponding formula where \7 f is replaced a closed 1-form

X. When X is not closed, there are a couple of extra terms in the calculation

which have a dX factor.

A Kahler version of this calculation of (6.113) was obtained by one of
the authors.
There is a computation analogous to (6.113) which one can perform for
solutions of a forward heat-type equation. In particular (see [287]).

LEMMA 6.99. If on a manifold Mn,

8

8t9ij = -2Rij,

~~ = Llf - R - IV' f 1

2

'

then
1
Z· iJ ·-=--;-R-iJ · - \7·\7 i J ·f + -g· 2t iJ ·

satisfies

(:t -LlL + 2\lf · \7) Zij = Yij - Zik ( Rjk + Y'j\lkf + ;t9jk)

- ( \7i\7kf +Rik+ 2 ~9ik) Zjk,

where

1

Yij ~ Y'iY'jR + 2\lkRijY'kf + 2RkijR.Y'kf\7.e.f + RikRjk + tRij·

One of the authors has made the following conjecture.

CONJECTURE 6.100. If (Mn,g (t)) and f (t) are solutions to the equa-

tions in Lemma 6.99, where g (t) has bounded nonnegative curvature opera-

tor, then

Yij ~ 0.