1547671870-The_Ricci_Flow__Chow

276 9. TYPE I SINGULARITIES

whence it follows that the limit

(9.13) E_ 00 ~ t->-00 lim E (h (t))

exists and is finite.
We want to use the compactness theorem to take a limit backwards
in time and get a new compact ancient solution of constant entropy. To do

this, we need a diameter bound. Since (N2, h ( t)) is an orientable positively-

curved surface such that

Rmax (t) ~ maxR (·, t)::::; C · ltl-l

N2

fort E (- oo, 0) , Klingenberg's Theorem (Theorem 5.9 of [27]) implies there
exists c1 > 0 independent of time such that

inj (N^2 , h (t)) 2: 7f 2: c1 /itl.

JRmax (t) /2

Then the area comparison theorem (§3.4 of [25]) implies there is c2 > 0

independent of time such that

4nx (N^2 ) · (w - t) =A (t);::: c 2 ·diam (N^2 , h (t)) · inj (N^2 , h (t))

2: c1c2 /ftl ·diam (N^2 , h (t)) ,

hence C3 < oo such that

(9.14) diam (N^2 , h (t)) ::::; C3 · vTtJ

for all t E (- oo, -1].

Now take any sequence of points and times (xi, ti) with

R (xi, ti) = Rmax (ti)

and ti ~ -oo. Consider the dilated solutions

hi(t) ~ R (xi, ti) · h (ti+ R (:i, ti)).

At each fixed x E N^2 , Corollary 9.21 implies that the function t f-7 R (x, t)

is nondecreasing in time. Thus the scalar curvature R of h satisfies

max R (x, t) = R (xi, ti)

N^2 x ( -oo,ti ]

and the scalar curvature R of hi satisfies

max R (x, t) = 1 = Ri (xi, 0).

N^2 x(-oo,0]

Then applying Klingenberg's Theorem again, we obtain the injectivity radius

estimate

inj (N^2 , hi (0)) 2: ~ = J2n.

v 1/2

By (9.14) and the Type I hypothesis, there is C 4 < oo giving the diameter

bound

diam(N^2 , hi(O)) = JR(xi,ti)diam(N2,h(ti))::::; C3JR(xi,ti) ltil::::; C4.