1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

1. A POINT PICKING METHOD 187

we are done.) Then we may define (xm+l, tm+i) E Ma to be a point such

that

(22.10)

(22.11)

1

0 < tm+l S tm and dg(tm+i)(xm+l, xo) S dg(tm)(xm, xo)+AI Rm 1-2 (xm, tm)·

By induction we also have that (22.10) and (22.11) hold with m replaced

by k E [1, m - 1]. Thus, for all 1 ::; k ::; m + 1 we have 0 < tk ::; ti ::; c^2 and

the curvatures at (xk, tk) are increasing at least geometrically:

(22.12)

so that

(22.13)

Hence, by (22.11) and (22.13), for all 1 ::; k ::; m + 1 the distance of Xk to
xo at time tk has the following upper bound:
1
dg(tk) (xk, xo) S dg(tk-i) (xk-i, xo) +Al Rm 1-2 (xk-i, tk-1)
1

S dg(ti) (x1, xo) +Al Rm 1-2 (x1, ti)

1

+···+Al Rm 1-2 (xk-i, tk-l)

(22.14) S co+ A ( 1 + t + · · · +

2

k~ 2 ) I Rm 1-~ (xi, ti).

1

Since (22.6) implies I Rm 1-2 (xi, ti) <co, we have

dg(tk) (xk, xo) < (2A + l)co

for 1 S k Sm+ 1. In particular, Xm+l E Bg(tm+i) (xo, (2A + l)co).

By (22.12), if the point (x, f) in the claim cannot be taken to be (xk, tk)

for any k E N, then we have

which contradicts the fact that the metrics g (t) are continuous in time so

that the space-time set UtE[O,e2] Bg(t) (xo, (2A + l)co) x {t} is compact and

IRml has a uniform upper bound on this set. Hence there exists£ EN such

that (x, f) can be taken to be (xe, tg). This completes the proof of Claim 1.

0

REMARK 22.4. From the proof of Claim 1 we see that it suffices to assume

that Bg(t) (x 0 , (2A + l)co) is compact in M for each t E [O, c^2 ] instead of

assuming that M is complete.