1547671870-The_Ricci_Flow__Chow

128 5. THE RICCI FLOW ON SURFACES

Let N be a constant to be determined, and set

<J? ~ tk+3 lvkRl2 + Ntk+2 lvk-l Rl2

Then there are constants a, bj, and Cj such that we can estimate

:t <J? ::::: L:\<J? + tk+^2 lvk Rl

2

[atR + (k + 3 - 2N)]

Lk/2J

+ Jtk+2 JVkRl2. L bjJtj+21vjRl2. tk-j+21vk-jRl2

j=l

L(k-1)/ 2J

+ N L CjJtk+l JVk-l Rl^2. tH^2 J'Vj Rl^2. tk-j+l JVk-j-lRJ^2

j=O

• N (k + 2) tk+l lvk-1 Rl2.

By the inductive hypothesis, there is a sufficiently large constant N depend-

ing only on g (0), and positive constants A, B, C, D depending on g (0) and

N such that

:t <J?::::: L:\<J? -Atk+2 l'VkRl2 + BVtk+2 JVkRl2 + c::::: L:\<J? + D.

As in Lemma 5.32, this suffices to yield the result. D

Now that our proof of Proposition 5.33 is complete, Theorem 5.28 follows

readily.

7. Strategy for the case that x (M^2 > 0)

We have established Theorem 5.1 for surfaces such that r :::; 0. In the

remainder of this chapter, we tackle the far more difficult case that r > 0.

For the convenience of the reader, we now summarize the path we will take

to prove that the normalized Ricci flow on a surface of positive Euler char-

acteristic converges exponentially to a metric of constant positive curvature.

7.1. The case that R (-, 0) 2:: 0. We first consider the special case

that the scalar curvature is initially nonnegative. By the strong maximum

principle, one then has Rmin (t) > 0 for any t > 0 unless R = 0 everywhere,

which is possible only if the initial manifold was a fiat torus. Hence (by

restarting the flow after some fixed time c > 0 has elapsed) we may assume

that R (., 0) > 0.

Recall that the trace-free part of the Hessian of the potential f of the

curvature is the tensor M defined in (5.9) as

1

M = 'V'Vf- 2L\f · g,

where by (5.8), one has L\f = R - r. In Section 3, we observed that the

tensor M vanishes identically on a Ricci soliton. And in Section 4, we saw

that the only self-similar solutions of the Ricci flow on a compact surface