1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

5. LOCALIZED NO LOCAL COLLAPSING THEOREM

such that

f(u) ~I

1 if u E (-oo, !e^1 -n],

0::; ¢/(u)::; 12exp (en+l + n) if u E (!e^1 -n, ~e^1 -n),

exp (e1-~-u) if u E [~el-n,el-n),

oo if u E [e^1 -n,oo).

For any A> 0 there exists C1 (n, A) < oo such that

(18.66) ¢~u) (¢'(u))

2

• ¢"(u) 2: (2A + nen)<P'(u) - C1(n,A)¢(u)

for u E (-oo, e^1 -n), that is,

(18.67)

ituJ (¢'(u))^2 - ¢"(u) - (2A + nen)<P'(u)

¢(u) 2: -C1(n,A).

To see this, define

¢(u) (¢'(u))^2 - ¢"(u) - (2A + nen)<P'(u)

C3 (n) ~ uE[~el-n,%el-n] min ¢(u) < oo.

On the other hand, for u E rne^1 -n, e^1 -n) we have

(18.68)

ituJ (¢'(u))^2 - ¢"(u) - (2A + nen)<P'(u)

¢(u)

1 2 2A + nen

• (el-n - u) (^4) (el-n - u) (^3) (el-n - u) 2'
  73
  For u E rnel-n, e^1 -n) the RHS of (18.68) is bounded from below by some

finite number -C 2 (n, A). Therefore (18.67) holds for u E (-oo, e^1 -n) with

-C1(n, A)~ min {-C2(n, A), C3 (n)}.

The next step is to apply the weak maximum principle to the function

h: M x [O, 1] --+ [-oo, oo] defined by

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

where L(w, T) = 4T£ (w, T).

Claim. Define

Then

(1)

where x is as in (18.60).

(2)

for T ::;! and w E M.

H (t) ~ min h (w, t).

wEM

H (1) = h (x, 1) = 2n + 1,

L(w, T) 2: -2n