1547671870-The_Ricci_Flow__Chow

120 5. THE RICCI FLOW ON SURFACES

5. Convergence when x (M^2 ) < 0

In this section, we will prove the following case of Theorem 5.1.

THEOREM 5.22. Let (M^2 , go) be a closed Riemannian surface with aver-

age scalar curvature r < 0. Then the unique solution g (t) of the normalized

Ricci flow with g (0) = go converges exponentially in any Ck-norm to a

smooth constant-curvature metric g 00 as t ---+ oo.

By Proposition 5.19, the solution g (t) exists for 0 < t < oo. By Proposi-

tion 5.15, the metrics g (t) are all uniformly equivalent. And by Proposition
5.18, there exists a constant C > 0 depending only on g 0 such that R is
exponentially approaching its average in the sense that

(5.19) IR -rl :::; cert.

So to prove the theorem, it will suffice to show that all derivatives of R are
dying exponentially.

LEMMA 5.23. On any solution (M^2 ,g(t)) of the normalized Ricci flow,

IV Rl^2 evolves by

(5.20) :t IVRl^2 = 6 IVRl^2 - 2 IVVRl^2 + (4R-3r) IVRl^2.

PROOF. Using (5.3) and the Ricci identity for a surface

1

V 6 = 6 V - -RV

2 '

we get
a I 3
at (VR) = V(6R+R(R- r)) = 6VR+ 2,RVR-rVR.

Hence

a 2 a ..

at 1v RI = at (lJViRVjR)

= (R-r) IVRl^2 + 2 \ 6VR+ ~RVR-rVR, vR).

Now the lemma follows from the fact that

6 IV Rl^2 = 2 (6 v R, v R) + 2 IVV Rl^2.

D

COROLLARY 5.24. If (M^2 , g (t)) is a solution of the normalized Ricci

flow such that r < 0, then there exists C 1 > 0 such that

(5.21) IV Rl^2 :S C1ert/^2.

PROOF. By (5.19), one has

:t IV Rl

2

:S 6 IV Rl

2

• 2 IVV Rl^2 + (r + 4Cert) IV Rl^2.