1547671870-The_Ricci_Flow__Chow

3. CURVATURE ESTIMATES USING RICCI SOLITONS 115

To compute the evolution equation for l\7 fl^2 , we use equation (5.10) and

recall that Rg = 2 Re on a surface, obtaining

at a IY'fl^2 =at a (.. lJ\7/vjf ) ( = atlJ a. ·) \i'if\i'jf + 2 ( at a \i'd ). \i'if

= (R - r) l\7 fl^2 + 2 (\7 !:::..f + r\7 f, \7 J)

= r l\7f1^2 + 2 (!:::.. \7 f, \7 J)

(5.14) = !:::.. l\7 fl^2 - 2 l\7\7 fl^2 + r l\7 f l^2 ·

Combining (5.13) and (5.14) gives the result, because

IMl^2 = l\7\7 f l^2 - ~ 2 (t:::..!)^2.

D

Applying the maximum principle to equation (5.12) yields a useful esti-

mate.

COROLLARY 5.17. On a solution of the normalized Ricci flow on a com-

pact surface, there exists a constant C depending only on the initial metric

such that

R- r :SH :S Cert.

Combining this result with Lemma 5.9 lets us estimate R both from

above and below.

PROPOSITION 5.18. For any solution (M^2 , g (t)) of the normalized Ricci

flow on a compact surface, there exists a constant C > 0 depending only on

the initial metric such that:

• If r < 0, then


• If r = 0, then

•If r > 0, then

C < R < C.

1 +Ct - -

In summary, we now have time-dependent upper and lower bounds for

the scalar curvature that are valid for as long as a solution exists. As we

shall see later in this volume, the long-time existence of solutions is a con-

sequence of these estimates. For now, we shall assume Corollary 7.2, which

tells us that long-time existence of solutions follows from appropriate a pri-

ori bounds on their curvature. To guide the reader's understanding, we

summarize the relevant arguments here.

STEP l. The smoothing properties of the Ricci flow (Theorem 7.1) reveal

that bounds on the curvature imply a priori bounds on all its derivatives