78 2. KAHLER-RICCI FLOW
REMARK 2.41 (1-dimensional normalized Kahler-Ricci flow). Recall the
following facts, due to Hamilton [180], about the normalized Ricci flow on
Riemannian surfaces gtg = (r - R) g. Throughout this remark, R and r
denote the Riemannian scalar curvature and its average, respectively. Note
that g (t) ~ g (~t) is a solution of the complex 1-dimensional normalized
Kahler-Ricci flow. The potential function f, defined by (2.32) and normal-
ized suitably by an additive constant, satisfies (see Lemma 5.12 on p. 113
of Volume One)
(2.38)af
at = !:if + r f.By the maximum principle, this implies If I ::; Cert. The gradient quantity
H ~ R-r + l\7 fl^2 satisfies (see Proposition 5.16 on p. 114 of Volume One)
(2.39)0
0~ = t:..H - 2 IMl^2 + rH,
where M ~ \7\7 f - ~!:if· g. By the maximum principle, we have (see Corol-
lary 5.17 on p. 115 of Volume One)
-Cert::; R-r::; H::; Cert.
This gives the exponential decay of IR-rl when r < 0. The norm squared
of the tensor M evolves by (see Corollary 5.35 on p. 130 of Volume One)
(2.40) %t IMl^2 = t:.. IMl^2 - 21v Ml^2 - 2R IMl^2.
Generalizing the 1-dimensional formula (2.38) to higher dimensions, the
potential satisfies a linear-type equation. (Strictly speaking, the equation is
not linear since the Laplacian is with respect to the evolving metric.)
LEMMA 2.42 (The potential f satisfies a linear-type equation). Under
the normalized Kahler-Ricci flow on a closed manifold Mn, the potential
function f, defined by (2.31) and normalized by an additive constant, satis-
fies(2.41) of = t:..f + ::._ f.
at n
PROOF. From (2.31) we computeOaB~ ( ~~) =! ( OaB~f) = :t ( R(X~ - ;g(X~)
- ro - ( r)
= oao~R - ;;, otga~ = oao~ !:if + ;;, f.
Since M is closed, it follows that
(2.42) ~~ =!:if+; f + c (t)
for some function c (t) and the lemma follows from the fact that we have the
freedom of adding a time-dependent constant in our choice of f ( x, t). D