1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

4. INTRODUCTION TO THE KAHLER-RICCI FLOW 79

COROLLARY 2.43 (Estimate for f). If Mn is closed, then, for the func-

tion f given by Lemma 2.42, we have

(2.43) Jfl :S Ge7/.

This is the first hint that the Kahler-Ricci fl.ow in the case where c 1 (M) <

0 is the easiest and that the case where c 1 (M) > 0 is the hardest.

We now compute for fin Lemma 2.42, for the normalized Kahler-Ricci

fl.ow, that

a 2 2 2 I

1

2 r 2

(2.44) ot JV' ail = Ll JV' afl - JV' a \7,sfl - \7aV'13f +;:;,JV' afl.

Define

h ~ Llf + JV' af J^2 = R - r + JV' afJ^2.

Similarly to (2.39), we have

LEMMA 2.44 (Ricci soliton gradient quantity evolution). For the nor-

malized Kahler-Ricci flow on a closed manifold Mn,

oh 2 r

(2.45) at = L:lh - JV' a V' ,af I + ;:;,h.

PROOF. We compute

ot a ( R - r) = L:l ( R - r) + I Rai3^12 - ;:;, r R

2 2r r^2 r

= Ll (R-r) + jV'aV'iJfl + -L:lf + - - -R

n n n

(2.46) = Ll (R-r) + IV'aV'iJfl

2

• ~ (R-r),
  n
  where we used (2.31) and (2.32). Equation (2.45) follows from summing this
  equation with (2.44). D

COROLLARY 2.45 (Estimate for R).

(2.47)

(2.48)

-Gen rt :SR - r :S Gen rt ,

JV' !12 :S Ge!iit.

PROOF. By (2.46), the lower bound for R - r follows from

(%t -Ll) (R-r) ~ ~ (R-r).

To get the upper bound for R-r, we observe that by the maximum principle,

we have rt

R - r :Sh :S Gen.

This also implies (2.48) since JV' f 12 = h - (R - r) :S h + Ge!iit. D

REMARK 2.46 (Exponential decay when c1 < 0). When c1 (M) < 0, so

that r < 0, (2.47) says that R approaches its average exponentially fast.

This suggests that the Kahler-Ricci fl.ow converges to a Kahler-Einstein

metric. Indeed, this is Theorem 2.50 below.