1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

96 19. GEOMETRIC PROPERTIES OF 11:-SOLUTIONS

Moreover, for all r E [(rr/2) + c-,oo) we have fc:(r)

r E (-oo, - (7r/2) - c-] we have fc: (r) = -1. We have

(19.18) f~ (r) = l f' (r + p) trJ (~) dp E [O, 1]

since f' E [O, 1]; in particular, for EE (0,7r/2),

f~ (0) = l cos ptfJ (~) dp > 0.

We also have

(19.19) !2 (r) = l !" (r + p) trJ (~) dp E [-1, OJ

1 and for all

for r 2: 0 (this uses the fact that f" (r) E [-1, O] for r 2: 0 and f" (r) is

odd).^18 Hence

(19.20) sup f~ (r) = f~ (0).

rE[O,oo)

Fix E > 0 sufficiently small. We consider the function

<p: [O, oo)---+ [o, J2 (n - 2)]

defined by

\D (r) * ../2 (n - 2)/, ( ../2 (n ~ 2Jfi (OJ).

(Note that 5n-l ( J2 (n - 2)) has Re= ~· Ifr 2: J2 (n - 2)f~ (0) (7r /2 + c-),

then <p (r) = J2 (n - 2).) Clearly,

(19.21) lim <p (r) = 0,
r--tO
(19.22) lim cp' (r) = 1.
r--tO
Since <p is the restriction to the nonnegative real numbers of a C^00 odd
function, we have both

(19.23) cp(r)=r+O(r^3 ) nearr=O

and

(19.24)

. d2k<p
r--tO hm d r^2 k (r) =^0

for all k E N. Hence

(19.25) <p^1 (r)=l+O(r^2 ) nearr=O.

Furthermore, by (19.20) we have

(19.26) sup <p^1 (r) = 1.

rE[O,oo)

(^18) Note that f (r) is C (^00) except at r = ±?T/2 and f' is a Lipschitz function on R