118 2. KAHLER-RICCI FLOW
Here we also used (2.33) and the following general identity:
. 1
(\J ,.y!:::.. - .6. \J 'Y) ha[3 = 2 \J p (\J 'Y \J p - \J p \J 'Y) ha[3
1
+ 2 ('V 'Y \JP - \JP \J 'Y) \J pha[3
1
= 2 \J p ( -Rrypauhu[3 + Rrypu[3hau)
1+ 2 ( -Rryu \J uha[3 - Rrypaif \J phu[3 + Rrypu{3 \J phau)
1(2.127) = 2 ( -\J ryRauhu[3 + \J ryRu[3hau)
1- Rrypaif \J phu[3 + Rrypu[3 \J phau - 2 Rryu \J uha[3'
where we used the second Bianchi identity (2.9). Simplifying (2.126), we
have
( :t -.6.) \J ryRa[3 = -~ (Ru[3\J ryRau +Rau \J ryRu[3 + Rryu \J uRa[3)
- Rrypcr{3\J pRau - Rrypaif \J pRu[3 + \J ry (Ra[3pJR,s-p) ·
Equation (2.125) now follows from this and the general formula
( :t -.6.) ('V ryRa[3X'Y) = [ ( :t -.6.) ('V ryRa[3)] X'Y + \J ryRa[3 (! -.6.) X'Y
- \J 8 \J ryRa[3 \J ,sX'Y - \Jo \J ryRa[3 \J 8X'Y.
D
EXERCISE 2.96. Prove Proposition 2.93 using Lemmas 2.94 and 2.95.10. Linear and interpolated differential Harnack estimates
In this section we consider a differential Harnack estimate related to the
estimate of H.-D. Cao considered in the previous section. This estimate has
applications in the study of the geometry and function theory of noncompact
Kahler manifolds with nonnegative bisectional curvature.
Let (Mn, g (t)), t E [O, T), be a complete noncompact solution of the
Kahler-Ricci flow with bounded nonnegative bisectional curvature. By Shi's
theorem, given an initial metric which is complete with bounded nonnegative
bisectional curvature, such a solution exists, at least for some short time
T > 0, with
c
l'V Rm (x, t) I S tl/ 2for some C < oo. Analogous to the Riemannian case (see Remark 2.25 in
this chapter or Theorem 10.46 on p. 415 of [111]), we consider a solution
to the linearized Kahler-Ricci flow. That is, we let ha[3 be a Hermitian
symmetric (1, 1)-tensor satisfying the Kahler-Lichnerowicz Laplacian heat
equation: