1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

no 2. KAHLER-RICCI FLOW

When Re > 0, the (1, 0)-vector minimizing the LHS of (2.113) is X^7 =

• (Rc-^1 )^7 ,o.7,oR, where (Rc-^1 t.B R 713 = <5~. Hence, if Re> 0, then (2.113)
  is equivalent to

8R + R _ (Rc-^1 )78 .7 R.7-R > O.

at t^7 0 -

Since R 78 :S Rg 7 8, we have - (Rc-^1 )^78 :S -~g7^8 , and hence

[) 2 [) 1 2

(2.114) 8tlog(tR)-l\.7log(tR)I = 8tlogR+ t - l\.'logRI ~ 0.

Without assuming Re > 0, we still obtain (2.114) by taking X7 = -~ \,77 R

in (2.113) and using Ra/3 :S Rga.B·

COROLLARY 2.89 (Integrated form of Kahler trace Harnack estimate).

If (Mn,g(t)) is a complete solution to the Kahler-Ricci flow with bounded

nonnegative bisectional curvature, then for any x1, x2 EM and 0 <ti < t2,

we have

R (x2, t2) > -e ti -1.6. 4

R(x1,t1)-t2 '

where Ll = Ll (x1, t1; x2, t2) ~ inf 7 ft~11 (t)l;(t) dt, and the infimum is taken

over all paths 'Y: [t1, t2] -t M with 'Y (t1) = x1 and'"'( (t2) = x2.

PROOF. By the fundamental theorem of calculus and (2.114), we have

for any 'Y: [t1, t2] -t M with 'Y (t1) = x1 and 'Y (t2) = x2,

t2R (x2, t2) ltz d

log R( ) = -d [logtR('Y(t),t)]dt

t1 x1, ti ti t

= 1:

2

[ ( :t log tR) ('"'! (t), t) +(.'log tR, 1 (t)) g(t)] dt

~ ltz [IVlogtRl;(t) + (\.'logtR,,-Y(t))g(t)] dt

ti

~ -~ 1t2 l'Y (t)l;(t) dt.
ti
The corollary follows from taking the infimum over all 'Y. D

For the normalized Kahler-Ricci flow, we have the following.

COROLLARY 2.90 (Integrated trace Harnack for normalized Kahler-Ricci

flow). If (Mn, g ( t)) is a solution to the normalized Kahler-Ricci flow on a

closed manifold with nonnegative bisectional curvature, then for any x1, x2 E

M and 0 < ti < t2,

(2.115)

where Ll is as above and r ~ 0 is the average (complex) scalar curvature.

REMARK 2.91. Note that l-e-~ti > e~ti_^1.

l-e-;:;:t2 - en:t2_1