1547671870-The_Ricci_Flow__Chow

124 5. THE RICCI FLOW ON SURFACES

I I I I I I I I I

,,,t-I -------

I I I I I I I I I I I

---,~

./

FIGURE 2. A surface of Euler characteristic zero

By Proposition 5.19, the solution g (t) exists for 0 < t < oo. By Propo-
sition 5.15, the metrics g (t) are all uniformly equivalent. To prove the the-
orem, we shall show that the scalar curvature and all its derivatives vanish
in the limit as t ----) oo.
We begin by deriving an estimate for the decay of the scalar curvature.

Recall from Section 3 that the potential of the curvature f satisfies

!::::.f =R

and is normalized by a function of time alone so that

8

a/= !::::.f.

LEMMA 5.29. Let (M^2 , g (t)) be a solution of the Ricci flow on a closed

surface with r = 0. Then there exists C < oo depending only on g (0) such

that for all t E [O, oo), the potential of the curvature satisfies

2 c

sup IV f (x, t)I :S -

1

-.

xEM2 + t

PROOF. It follows from Proposition 5.16 that

:t IV fl^2 = 1::::.1v f l^2 - 21vv fl^2.

By the maximum principle, there is Co= Co (g (0)) such that IV f l^2 :::; Co for
all time. To improve this estimate, we apply another BBS technique. (The
method of obtaining gradient bounds via maximum-principle arguments will
be discussed further in Chapter 7. Please see the references cited therein.)
Noticing that

and

:t (t IV f l

2

) :::; !::::. (t IV 11

2

) +IV! 1

2

%/

2

= !::::. (!

2

) - 2 IVfl

2

'