1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

58 18. GEOMETRIC TOOLS AND POINT PICKING METHODS

PROOF. We shall construct by recursion a finite sequence of points
{(yi, ti)} which ends with the desired point (y, t).
STEP 1. Construction of a finite sequence of points. Start with (y1, ti)
and suppose that we have constructed points (Yi, ti) E NB,C for 1 :::; i :::; k,
where k;:::: 1, which satisfy for 1 :::; i:::; k - 1,

(1)

(2)

(18.31)

(3)

(18.32)

If the curvature bound (18.29) holds for (y, t) = (Yk, tk) and all (y, t) E
NB,C satisfying (18.30), then we stop the construction of the sequence.
Otherwise, we may choose (Yk+li tk+l) so that
(i) (Yk+l, tk+l) E NB,C,
(ii)

(18.33) tk+l E (tk - AR-^1 (yk, tk), tk],

(iii)

(18.34)

(iv)

(18.35)

Clearly (Yk+li tk+i) satisfies (1), (2), and (3).

In this way we obtain a sequence {(yi, ti)}f= 1 , where a E [1, oo]. We now

proceed to prove that this sequence must be finite, i.e., a < oo.

By (18.32) and (18.27), we have

(18.36) R (yk, tk) > 2k-l.R (y1, ti) > 2k-l ( C + B(t1 - to)-^1 )

for k < a+ 1.^10 In particular, if a= oo, then limk-too R (Yk, tk) = oo.
Using (18.27), (18.31), and (18.36), we estimate

(18.37)

dh(tk) (YkiP) :::; dh(tk-i) (Yk-1,P) + A^112 R-^1 /^2 (Yk-li tk-1)

k-1

:::; dh(t1) (y1,p) + L A^112 R-^1 /^2 (Yi, ti)

i=l

1 4A^112

< -+ --------

4 (C + B(t1 - to)-1 )i/2.

(^10) By convention, oo + 1 ~ oo.