1547671870-The_Ricci_Flow__Chow

15. AN ALTERNATIVE STRATEGY FOR THE CASE x (M^2 > 0) 167

denote the area of M;. Since 'Y is a geodesic, the Gauss-Bonnet formula
(5.51) reduces to

27f = x (M;) = ( K dA

}M2

for i = 1,2. This implies that Kmax · min{A1,A2} ' 2:: 27r, equivalently that

{

max -,-^1 1 } <--. Kmax

A1 A2 - 27f

Thus by (5.47), we get the estimate

CH (M2' g) :S CH ('Y) = L ("f )2 ( ~1 + ~2) :S K:ax L ("f )2'

which we write in the form

L ('Y)^2 2:: K7r CH (M^2 ,g).

max

If Lmin denotes the length of the shortest geodesic loop in (M^2 , g), then
Klingenberg's result says

[inj (M2' g) J 2 2:: min { K:ax' L~in}.

Since we have CH (M^2 , g) :::; 47r by Lemma 5.84, we conclude that

2 2 { 7f^2 CH(M^2 ,g) 7f 2 }

[inj (M ,g)J 2:: min Kmax. 47f ' 4Kmax CH (M ,g).

D

Combining this result with Theorem 5.88, one obtains the following:

COROLLARY 5.95. If (M^2 ,g(t)) is a solution of the Ricci flow on a

topological S^2 with Gaussian curvature bounded above by K max ( t), then

[inj (M

2

,g(t))]

2

2:: 4 Km:(t)CH (M

2

,g(O)).

Our next task is to extract a limit of dilations about the singularity.

Since the singularity time is T < oo, the solution (M^2 , g ( t)) exhibits either

a Type I singularity

sup IR(-, t)I (T - t) < oo

M^2 x[O,T)

or Type Ila singularity

sup IR(-,t)l(T-t)=oo.

M^2 x[O,T)

(See Section 1 of Chapter 8.) In the first case, we take a sequence of points

Xj E M^2 and times tj / T chosen to give a Type I limit, as in Section 4.1

of Chapter 8; in the second case, we chose Xj E M^2 and tj / T to give a

Type II limit, as in Section 4.2 of Chapter 8. In either case, we ensure that