1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

5. EXISTENCE AND CONVERGENCE 83

If r =/= 0, the integral on the RHS is equal to C~n ( e;;t - 1). When r = 0, we

obtain Cit.

Similarly for the lower bound. D

REMARK 2.54. The qualitative dependence of the estimate (2.56) on the

sign of r should be compared to Corollaries 2.43 and 2.45.

The next estimate, a bound for the determinant of the complex Hessian

of cp, is also a straightforward application of the previous result and the

estimates (2.55) and (2.56) to equation (2.50).

LEMMA 2.55 (Estimates for the volume form-uniform when r :S 0). If

r :SO, then there exists a constant C 2: 1 such that

1 det (g 0 J3(x, t))

(2 57) - <. :S c

· C - det (9ajj(x, 0))

for all x EM· and t 2: 0. If r > 0, then

_en (efi:t_1) det (9ajj(X, t)) Gn (efi:t_1)

e r < :Ser

• det (9ajj(x,O))
  for all x EM and t 2: 0.
  PROOF. .Applying (2.47) to (2.35), we have

a det (9ajj(X, t)) rt

(2.58) - log =Ir - RI :S Geri.

at det (9ajj(X, 0))

Hence, if r =/= 0, then

det (9ajj(x, t)) Cn ( J:_t )

log :S - en - 1 '

det(gajj(x,O)) r

so that
_ Gn (efi:t-1) det (9ajj(X, t)) Gn (efi:t-1)

e r < :Ser.

• det (9ajj(x, 0))

In particular, if r < 0, then

e ~n < det (9ajj(x, t)) :S e-c;.

• det (9ajj(x, 0))

When r = 0, equation (2.58) is not strong enough to uniformly estimate
det(g °' (3-(x ' t)). In this. case we use
det(g°'i3 (x,O))
a det (9ajj(x, t)) af
-log =r-R=-llf=--,

at det (9ajj(X, 0)) at

which implies

det (9ajj(x, t))

log d ( -( O)) = -f (x, t) + f (x, 0).

et 9af3 x,