1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

122 2. KAHLER-RICCI FLOW

In each of the above instances, we also have the complex conjugate equations;
we leave it to the reader as an exercise to write these down. Recall that Cao's
Kahler matrix differential Harnack quadratic is (2.111):

Rrx!3

z rx!3 = b.RCi73 + Rrxf31J Ryi5 + V' ,RCi73 V1 + V' 1Rrx73 v, + Rrxf31J V1 Vo + -t-.

From (2.132) and (2.133) and their conjugate equations, while substituting

in (2.134), (2.135), (2.136) and their conjugate equations, we obtain

(8t §-8.) Z = Z CitJ 1-ih-r:i CitJ +h /u <(V' Ci V,-"( -R Ci/ - - !g t Ci/ -) (V'-V.-R-· Ci u Ciu - !g-·) t Ciu

(2.137) -t 1 ( -h-^2 (H +en) )

1 5(\i'5V^1 +V'^1 Vo)+2R5^1 h 1 5+ t +2eR

+ h 1 5\i' a V1V'°'1/8.

From (2.134) and the trace of (2.136), we have

Z = R 1-ih-r:i - !h, 7-iY'-Vr:i - !hr:i-\i' Vr.i + H +en +eR.

°'" °'" 2 °'" Ci fJ 2 "°' Ci fJ t

Substituting this into the RHS of (2.137) yields

Z°'73ha(3 + h 15 ( V' rx V1 - Rrx1 - t9rx')i) ( V' a Vo - Rai5 - t9ai5) ~ 0.

Hence we have for the minimizer V satisfying (2.134)

(%t -8.) (t

2

z) ~ o.

By applying the maximum principle (see pp. 639-640 of [290] for details),
we may conclude that t^2 Z ~ 0 for all t > 0.
We have the following matrix differential Harnack estimate due to one
of the authors [287].

THEOREM 2.101 (Matrix interpolated differential Harnack estimate).

Let (Mn, g (t)) be a complete solution of thee-speed Kahler-Ricci flow

8

(2.138) Bt9rx(3 = -eR°'73'

where e > O, with bounded nonnegative bisectional curvature, and let u be a

positive solution of the forward conjugate heat equation

8u
(2.139) Bt = b.u + eRu.

Then for any (1, 0)-form V we have

(2.140)

Equivalently, f ~ log u satisfies

1
(2.141) f rx(3 + eRrxf3 + t9rx(3 ~ 0.