1547671870-The_Ricci_Flow__Chow

170 5. THE RICCI FLOW ON SURFACES

PROPOSITION 5.97. Let (M^2 ,g (t)) be a solution of the Ricci flow on

a topological 2-sphere with g (0) = go. Then g (t) exists for 0 :S t < T ~

1/f, where f denotes the average scalar curvature of the metric go, and the
singularity at time T is of Type I.

PROOF. Suppose the singularity is of Type Ila. Since

A (t) =A (0) - 87rt = 87f ( ~ - t) = 87f (T - t)

by (5.53) and (5.54), this happens if and only if

sup Rmax (t) ·A (t) = oo.

tE[O,T)

Since

Area (M^2 , gj (0)) = R (xj, tj) ·A (tj) = 87rR (xj, tj) · (T - tj) ---too

by (5.55), the limit (M~,g 00 (0)) has infinite area. In particular, M~ is
noncom pact, and g 00 ( t) has infinite area for all times t E ( - oo, oo). Since
we have
Jim min R ( ·, f) / 0,
t-too M^2
hence
lim minR (-, t) / 0
t-tT M2
by Lemma 5.9, the scalar curvature of (M~, g 00 (0)) is nonnegative, hence
strictly positive by the strong maximum principle. Moreover, R~ E (0, 1]
attains its maximum at (x 00 ,0). By Lemma 5.96, (M~,g 00 (t)) is the cigar
soliton.
As we observed in Section 2 of Chapter 2, the cigar is the metric given
in polar coordinates (p, e) on IR^2 by
1 2 p2 2
g"E = -1--2dP + -1--2de.

+p +p

As we saw in Section 2 of Chapter 2, it attains its maximum curvature at

the origin and is asymptotic to a cylinder as p ---t oo. In particular, for any

c; > 0 there is C » 1 large enough so that the circle p = C has length

27r - c; < L < 27f and the open disc p < C has area A > 1/c:. Hence

if (M^2 , gj (t), Xj) is a sequence of closed pointed manifolds converging to

the cigar (IR^2 , g"E, 0), it is easy to see that there is j E N so large that

CH (M^2 , gj) < c:. Since this contradicts Theorem 5.88, we conclude that

the Type Ila singularity is impossible. 0

COROLLARY 5.98. The scalar curvature of any solution (M^2 , g ([)) of

the normalized Ricci flow on a topological 2-sphere is uniformly bounded.

PROOF. Since

g ( t) = ( 1 - rt) g ( f) = r ( T - t) g ( f)