1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

124 2. KAHLER-RICCI FLOW

Hence

(

8 ). :

Bt - b.L Sa/3 2: fa,rff31 + \17f\lrySa/3 + \l~f\11Sa/3

• ~Sa1 (ff3ry - c:Rry/3 - ~gry/3)

+ 2 1 (. fa1 - c:Ra7 - tga7^1 ) S13ry·

The estimate (2.141) follows from an application of the maximum principle;
see [287] for details. D

Tracing (2.140), i.e., multiplying by ga/3 and summing, we have

COROLLARY 2.102 (Trace interpolated differential Harnack estimate).

Under the hypotheses of Theorem 2.101,

nu -

b.u + c:uR + t + gaf3 ( Ua V13 + u13 Va + u Va V13) 2: 0,

which, by taking Va = -~ and then dividing the resulting expression by u,

implies the equivalent inequality

n

b. log u + c:R + -t > - 0.

Let N ~JM ulogudμ be the (classical) entropy of u. We have under

(2.139), gtdμ = -c:Rdμ (since gt detg'YJ = -c:RdetgryJ), and

. d: =JM (b.u + c:Ru + (logu) b.u) dμ

=JM (b.logu + c:R) udμ

2: -~ r udμ.

. }M

In other words,

(2.143)

11. Notes and commentary

Some books containing material on or devoted to complex manifolds and

Kahler geometry, in essentially chronological order, are Weil [370], Chern

[95], Goldberg [157], Kobayashi and Nomizu [236], Morrow and Kodaira

[275], Griffiths and Harris [166], Aubin [13], Kodaira [238], Besse [27], Siu

[334], Mok [270], Tian [347], Wells [371], and Zheng [383]. We refer the

reader to these books for the proper study of Kahler geometry.

For the Ricci flow on real 2-dimensional orbifolds, see L.-F. Wu [372] and

[112]. For the Ricci flow on noncompact Riemannian surfaces, see Wu [373],