1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

5. EXISTENCE AND CONVERGENCE 89

on the time interval [O, T). Once we have a uniform C^2 '°'-estimate of <p on

[O, T) (we shall prove the C^2 '°'-estimate in the next subsection), by choosing

a time to < T close enough to T, the NKRF (2.50) with initial condition

<p (to) can be solved on [to, to+ c], where c depends on the C^2 '°'-estimate of <p

but not to. If we choose to = T - ~, this implies that we have a solution <p ( t)

fort E [O, T + ~]. This contradicts T being the maximal time of existence;

hence the theorem is proved. D

We shall supply the details for the C^2 '°'-estimate in the next subsection.

5.3. The C^2 '°'-estimate of <p. We now proceed to derive the C^2 '°'-

estimate for <p, which by (2.50) and (2.54) is a solution to the complex

Monge-Ampere equation:

log det ( g~,6 + <p a,6) = h + log det g~,6 ~ h,

·and

(2.77)

• r
  h(x, t) = -f(x, t) + f(x, 0) - -<p(x, t).
  n
  By (2.70), there exists a constant A> 0 such that the Kahler metric 9a,6 ~
  g~,6 + <p a,6 satisfies

(2.78)

By the fact that the bounded function f satisfies flf = R - r and that

we have the scalar curvature bound (2.47), standard Lq theory (which only

requires the uniform boundedness of the coefficient matrix ( g°',6) from above

and below) implies that llfllao(M) + llVV fllLq(M) is bounded and hence by

(2.77), llhlloo(M) + llVVhllLq(M) is bounded (uniformly int when c1 (M) :::::;

0) for any q < oo (independent oft).

Note that h is globally defined whereas h is only locally defined. How-

ever, for compactly contained open subsets U of a holomorphic coordinate

chart of go, llhllaocu)+ llVVhllLq(u) and llhllao(u) + llVVhllLq(U) are equiva-

lent. Let B(R) denote the Euclidean ball of radius R centered at the origin

in en. Since M is compact, there exists a finite collection of open sets

{Uk}t'~ 1 and normal'holomorphic coordinates Zk = {zf}:=l defined on Uk

(independent oft) such that

No

B (3Rko) c Zk (Uk) and LJ zJ;^1 (B (Rko)) = M,

k=l

where Rko > 0. Hence it suffices to prove the C^2 '°'-estimate for <p in each

open set zJ;^1 (B (Rko)) assuming that

(2.79) llhllaocuk) + llVVhllLq(Uk) :S C < oo.

From now on we work in a fixed coordinate chart. More precisely, we

use Zk to push forward our discussion to the Euclidean ball B ( 3Rko). For