1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

5. LOCALIZED NO LOCAL COLLAPSING THEOREM 71

(here we used 11,^1 1nr^2 < c1 (n) :::; !). From this and (18.62), we conclude that

there exists 11, (n, A) > 0 such that 11, 2: 11, (n, A). Theorem 18.36 is proved,

except that we need to give a proof of (18.63).

We shall need the following lemma, which will be proved in the next

subsection, to estimate V (1) from below.

LEMMA 18.38. Let (Mn,g(t)), t E [0,1], be a complete solution of the

Ricci flow with bounded curvature. Suppose for some xo E M that we have

(18.65) IRml :::; 1 in Bg(o)(xo, 1) x [O, l].

Then

(i) JWJg(t) :::; en-^1 IWJg(O) for any z E Bg(o)(xo, 1), t E [O, l], and

WETzM;

(ii) Bg(t)(xo, e^1 -n) C Bg(o)(xo, 1) for all t E [O, 1];

(iii) there exists c4(n, A) < oo such that the reduced distance, as defined

above, satisfies

Next we give an upper bound (depending on n and A) of £(q, 1) for

q E Bg(l)(xo, 1) = Bg(o)(xo, 1). By Lemma 18.38(iii), there exists x1 E

Bg(~)(xo, e^1 -n) such that

Let 'Yl : [O, !J ---+ M be a minimal £-geodesic with (/'1 (r), r) joining (x, 0)

to (x 1 , !) , where x is as in (18.60). By Lemma 18.38(ii), for any q E

Bg(i)(x 0 , 1) there exists a minimal (Riemannian) geodesic

joining x 1 to q, with respect to the metric g(l), which satisfies

By Lemma 18.38(i), we have

I

, .. / 12 (s)[: g(s) < - e2(n-1) I"/ 12 (s)I: g(l) -< 16e2(n-1)_