1547671870-The_Ricci_Flow__Chow

NOTES AND COMMENTARY 251

defined as before. Each solution (g 00 )i exists on the dilated time interval

(- oo, (w - ti) IRmoo (xi, ti)I)

which contains the subinterval [-ai, wi] on which we have a good estimate of

the curvature. Indeed, (8.28) implies that for all x E M~ and t E [-ai, wi],
one has

l(Rm oo)i (x, t)I = IRmoo l(xi, ti)I I Rm oo ( x, ti+ I Rm oo t(xi, ti)I) I

< (ti - Ti) ltil

• (1 - fi) (ti+ IRm=t(xi,ti)I - Ti) lti + IRm=t(x;,ti)l I

(1 - Ei) (ai + t) (wi - t)'

Because ai ---+ oo, wi ---+ oo, and fi "\, 0, we conclude that the pointed limit

solution (M2 00 , g2 00 (t), x2 00 ) is an eternal solution that satisfies

sup IRm2ool :S 1 = IRm200 (x200,0)I.

M-q-= x ( -oo,oo)

N ates and commentary

The main reference for this chapter is Section 16 of [63]. Limit solutions

are obtained by the application of the Compactness Theorem we reviewed

in Section 3 of Chapter 7. Limits formed from dilations about sequences of

points and times that are not curvature essential will be considered in the

next volume.