1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

282 6. ENTROPY AND NO LOCAL COLLAPSING

for any function¢ of space and time. We may also rewrite the evolution of
Re in the following funny way:

(~ at + ~L) R-· iJ -- 2~LD .LL</,J· ..

Hence

( :t + ~L) ( \Ji\Jjf + Rij + 2 :9ij)

(

(^2) ) ( 8 e) c
2
=2~LRij+\Ji\Jj -R+l\Jfl -2 8trij \Jd-2r29ij--:;.Rij c
= 2 ( ~LRij - ~\Ji\JjR + Rkije\Jkf\Jd)

• 2\Jk \Ji\Jjf\Jkf + 2\Ji\Jkf\Jj\Jkf
  c^2 c


• 2 (\JiRje + \JjRie - \Je~j) \Jd - 2 T 2 9ij - -Rij T
  = 2\Jk ( \Ji\Jjf + Rij +
  2
  c
  7
  9ij) \Jkf


• 2 ( ~LRij - ~\Ji\JjR + ~kRjk + 2 : ~j)


• 2 ((Piej + Pjei) \Jd + Rkije\Jkf\Jd)
  2c c^2

+ 2\JiY'kf\Jj\Jkf - 2RikRjk - -Rij - -

2 2

9ij

T T

and the result follows from rewriting the last line. D

REMARK 6.96. In the evolution equation for f, we may replace -~~ on

the RHS by any function of t.

Since the matrix Harnack quadratic is the space-time Riemann curva-

ture, we ask the following.

and

PROBLEM 6.97. Is there a space-time interpretation of equation (6.113)?

For convenience, we define

T~-iJ = • .L D LiJ .. - \J·\J i J ·J + ~g·· 2r iJ

1 c

H (X)ij ~ ~L~j - 2 \Ji\JjR + RikRjk + 27 ~j

+ (Piej + Pjei) Xe+ RkijeXkXe.

Then we may rewrite (6.113) as

( :t + ~L - 2\J f · \J) Sfj = 2H (\J f)ij - SfkTfk - SjkYi\,

where Sfk is defined in (6.27).