1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

94 2. KAHLER-RICCI FLOW

Putting these together, we obtain
1

w1 (2R) ~ (Vol ~(R) k(R) (Mi (2) - m1 (2) )Pr

1

:<; (Vol ~(R) k(R/ W1 (y) - m1 (2) )Pr

1

• ( Vol~(R) k(R/M1(2) -w1(Y))Pr

:S; C ( w(2R) - w(R) + RllV'hllco + R

2

<q;n) llV'VhllLq).

Since there is nothing special about the index 1, summing the corresponding

upper bounds for wv(2R) implies

w(2R) :S; C ( w(2R) - w(R) + RllV'hllco + R

2

<q;n) llV'VhllLq)

with a different constant C < oo. We conclude the following.

LEMMA 2.66 (Oscillation estimate). There exists 5 < 1 (i.e., 5 = 1 - ty)

such that for any R::; Ro we have on B(3Ro),

2(q-n) _

(2.94) w(R)::; 5. w(2R) + RllV'hllco + R-q-llV'V'hilLq·

2(q-n) _

Now since R-q-llV'V'hliLq(U) and llV'hllco(u) are bounded by (2.79),

the Holder continuity of V'Vu on B(Ro) can be derived from (2.94) by a

standard argument; see Moser [276], or Corollary 4.18 on p. 91 and Lemma

4.19 on p. 92 of Han and Lin [195], for example. Finally, the Holder

continuity of \7Vu is equivalent to the Holder continuity of V'Vr.p.

5.4. Proof of Theorem 2.51. Finally, we give the proof of Theo-

rem 2.51, i.e., the proof of the convergence of the normalized Kahler-Ricci

flow in the case where c1 < 0. Assume, without loss of generality, that

r = -n, which can be achieved by scaling the initial metric go. Notice that

we have that f = -~ satisfies

f)

fJtf = !::.g(t)f - f

and lf(x, t)i ::; ce-t and IV' JI (x, t) ::; ce-t. (The last inequality is by

(2.48).) That is,

for some C1 < oo.

Now b.. 9 (t) = g0/.i3 az~~z/3 and g0/.i3 = g~i3 + r.p0/.i3· So the C^2 ,0l.-estimate for

<p implies a COl.-estimate for the coefficients g0/.i3. Thus we may apply the