1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

74 18. GEOMETRIC TOOLS AND POINT PICKING METHODS

(3) There exists y E B 9 (~) (xo, e^1 -n) such that

H(~)=h(y,~).

(4) At any point (y, t) where h(y, t) = H (t) with t 2: ~' we have

(18.69) (:t -~) h(y,t) 2:-(2n+C1(n,A))h(y,t).

Assuming the claim, we finish the proof of part (iii) of Lemma 18.38. It

follows from part (4) of the claim and Lemma 3.5 in [89] (on differentiating

a minimum function; see subsection 1.1 of this chapter) that

d

dt H (t) 2: - (2n + 01 (n, A)) H (t) fort 2: ~·

Integrating this on [~, 1] while using H (1) = 2n + 1, we obtain

H(~) ::; c5 (n, A).

This and part (3) of the claim imply that there exists y E B 9 (~)(x 0 , e^1 -n)

such that

c5 (n, A) 2: h (y, ~)

= ¢(d 9 (~)(xo, y)) · (2C(y, ~) + 2n + 1)

2:2£(y,~)+2n+l

whenever C(y, ~) 2: -n - ~ (since ¢ 2: 1). Thus we have (iii).

Finally we give a proof of the claim.

(1) This part follows from dg(l) (x, xo) ::; A (by (18.60)) and L(w, 0) =
d~(O) (w,x), which is justified by the equation after (7.94) in Part I. Indeed,
we have

h(w, 1) = ¢(dg(l)(xo, w) -A)· (L(w, 0) + 2n + 1)

= ¢(d 9 (1)(xo, w) - A)· (d~(o) (w, x) + 2n + 1)

2: 2n + 1

with equality when w = x since ¢ ( u) = 1 when u ::; 0.

(2) Let Rmin (t) ~ minzEM R (z, t). From the evolution equation for the
scalar curvature, we have fftRmin 2: ~R!in' so that