1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

5. ROTATIONALLY SYMMETRIC EXPANDING SOLITONS 31

Hence W^2 / X 2::: c1 for some positive c1, for r sufficiently large. Thus, the

denominator in the integral (1.61) is bounded below,

.w2 -x2 > >.w2 - w4

• (c1)^2 '
  and it follows that r = O(l/W).
  The orbital sectional curvature satisfies
  Y^2 - ( n - 1 )X^2 X^2 W^2
  V1 = >---> ---

n(n - l)W^2 nW^2 - n(c1)^2

and so decays at least as fast as 1 / r^2. The equation

(1.^62 ) ( 1)

(X^2 - aX + >.W^2 )

Vz + n - VI = - wz

shows that in order for v2 to also decay to zero, it is necessary that

(1.63) lim X/W^2 = >.yin.

s-++oo

We already know that X/W^2 is bounded above, and we can then apply the

Cauchy mean value theorem (as in Lemma 1.33) to the equation

!ix :2 (x^2 - >.w^2 + ar -1) +>.yin

2 ds _ -------------

ds d W2 - x2-;.w2

to obtain the desired limit. Next, dividing (1.62) by W^2 gives

-v2 v1 X^2 >. - a:2

w2 = (n-l)wz + W4 + w2.

In order to show that v2 decays at least as fast as 1/r^2 , we need only show

that the last term is bounded. This follows from the differential equation

ds d (>.-a~) w2w =(-1+2>.w2_3x2) (>.-aK) w2w2

x 2 y2

+>.(X-a)wz -a W4'

where the boundedness of the terms on the second line follows from the limit

(1.63) and the negative curvature condition Y^2 :::; (n - l)X^2.

To establish the last assertion, it suffices to show that dw / dr = x =
v:n=1 X/Y is bounded above and below for r sufficiently large. First,
because of (1.62) we can say that

d ( w2) z z w2 z w2

ds y =(X +aX->.W)-y:S:2X-y.

Because of the limit (1.63),

d w^2

-log-< ds Y - CW^4