1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

6. NOTES AND COMMENTARY

Next we compute that

JM IR - rl dμ ~JM ((R-Rmin) + (r - Rmin)) dμ = 2A (r - Rmin),

where A= Vol (g(t)). Thus, from (32.21), we have

1

00

JM IR - rl dμdt ~ 2A 1

00

(r (t) - Rmin (t)) dt ~ 2AC.

Hence

211

l

t+l! IR9oo(r) + 61 dμoodT ~ ,lim lt+l! IR9,(r) - r9;(r) I dμ9,(r)dT = 0

t M 00 i-+oo t M

for all t E R Therefore R 900 (x , t) = -6 for all x E M 00 and t E ffi.. From

8 2 2 2

0 = at R9oo = 6.R9oo + 2 IRc 900 I - 3rooR9 00 = 2 IRc 900 + 2gool '

we conclude that

(32.23) Re 900 = -2goo.

Since dim(M 00 ) = 3, this implies that

sect (g 00 (t)) = -1.

Finally, note that Vol (g 00 (t)) ~A from C^00 Cheeger- Gromov convergence. D

By Proposition 32.12, if the limit so lution (M 00 , g 00 (t)) is on a compact man-

ifold, then M 00 ~ M and so M admits a metric with constant negative sectional

curvature -1. This is case (B4) in Subsection 2.2 of this chapter.

The next chapter is devoted to the analysis of the most difficult case where

M~ is noncompact and to finishing the proof of Theorem 32.2 by showing that

case (B5) must hold.

6. Notes and commentary

For work on nonsingular solutions in dimension 4, see Fang, Y. Zhang, and
Z. Zhang [106].
For the finiteness of the number of surgeries of the Ricci flow on 3-manifolds,
see Bamler [19], [20], [21], [22], [23].
Definition 32. 1 is Definition 1.1 in [143].
Theorem 32.2 is Theorem 1.3 of Hamilton [143].
For (32.3), see Corollary 6.64(5) in Volume One.
For Lemma 32.3, see Theorem 2.1 in [143].
Subsection 4.1 follows §5 of [143].
Subsection 4.2 follows §6 of [143].
Proposition 32. 12 is Lemma 7.1 in [143].