1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

84 2. KAHLER-RICCI FLOW

By the uniform bound (2.43) on f, we conclude

for some C 2:: i.

1 det (ga13(x, t)) -

-=< <C

C - det (gajJ(x, 0)) -

D

REMARK 2.56. Alternatively, when r i= 0, applying the estimates (2.55)

ai:id (2.56) to equation (2.50), we have.

det 9a13(x, t) I ocp r I

log det gajJ(x, 0) = ot (x, t) - -:;;,cp(x, t) + f(x, 0)

S llJ(·, O)lloo ( ef;t + lef;t -11+1),

since max{C1, C2} S II!(·, O)lloo· In particular, if r < 0, then

e-211!(·,0)lloo < det (gajJ(x, t)) < e2llf(·,O)lloo.

• det (gafJ(x, 0)) -

Lemma 2.55 is the first step towards proving that \7 a \7 jJ'P is bounded

and that gajJ(x, t) is equivalent to gajJ(x, 0), which in particular implies_ that

gajJ(x, t) is always positive definite.

Next we estimate the trace of gajJ(x, t) with respect to gajJ(x, 0). Let

(2.59) Y(x, t) ~ ga/J(x, O)gajJ(x, t)

be the trace-type quantity we want to estimate. As we shall see below, a

bound for y (t) will imply a C^2 -estimate for cp (t). From (2.52) we have

y = n + f:l.g(O)'P,

(2.60) n = ga!3(t)gajJ(O) + l:l. 9 (t)'P·

Hence an estimate for Y implies an estimate for l:l. 9 (o)'P· Let Aa denote the

eigenvalues of gajJ(t) with respect to gajJ(O). Then

(2.61) Y = L~=l Aa

and the eigenvalues of ( a::Jf 213 ) with respect to gajJ(O) are Aa - l. If Y S C,

then as long as gajJ(t) is positive-definite, Aa SC for each a. On the other

hand, by Lemma 2.55, we have

1 n

CS II Aa SC,

O!=l

where for r S 0, the constant C is independent of time, whereas for r > 0, C

depends on time but remains bounded as long as the solution exists (though

the bound may tend to oo as time approaches oo). Hence there exists a

constant c > 0 such that Aa 2:: c, where c is independent of time for r S 0

and may depend on time for r > 0. So indeed, gajJ(t) remains positive-

definite as long as the solution exists and we have c S Aa S C', for some