70 2. KAHLER-RICCI FLOW
Since the Kahler-Ricci flow is a heat-type equation for Kahler metrics,
some evolution equations we shall derive later in this chapter use the follow-
ing commutator formula. If a is a time-dependent (0, 1)-form, then under
the Kahler-Ricci flow,
[{) ] 1 1ot - ~' \7 a a13 = 2 \7 aR 8 13a3 - 2 RaJ \7 oa13 + Ra:yo/3 \7 1 a3.
Another basic formula is the commutator of the heat operator and the Hes-
sian. First, by (2.19) and (2:20), we have for any function f on M,
\7a\713~f = ~L \7a\713f
1 1
(2.22) ~ ~ \7a\713f + Ra/3o;y \7 'Y \73f - 2RaJ'Vo\713f - 2R'Y13\7 a 'V;yf,where ~L is called the (complex) Lichnerowicz Laplacian. If f is a
function also of time, then (2.22) tells us
(2.23) 'Va\713 (%t -~) f = (:t -~L) 'Va\713f.
Note that since \7a\713 = 8a[J(:J acting on functions, we have gt (Va \713) = 0
when acting on functions.
REMARK 2.25. Formula (2.23) has the following Riemannian analogue.For a time-dependent function f and under the Ricci flow, we have
(2.24) \7i\7j (! -~) f = (:t -~L) \7i\7jf,
where ~L, defined by
~LVij ~ ~Vij + 2Rkij£Vk£ - RikVjk - RjkVik,
is the (Riemannian) Lichnerowicz Laplacian acting on symmetric 2-tensors
(see Lemma 2.33 on p. 110 of [111]). In fact, (2.24) is a special case of
(2.23).
3. Existence of Kahler-Einstein m~trics
In this section we discuss what is known about the existence and unique-
ness of Kahler-Einstein metrics, which are canonical (constant Ricci curva-
ture) metrics on Kahler manifolds. The Kahler-Einstein equation is elliptic
whereas the Kahler-Ricci flow, discussed beginning in the next section, may
be considered as its parabolic analogue.
First recall the following 88-Lemma, which is a consequence of the
Hodge decomposition theorem.
LEMMA 2.26 (d-exact real (1, 1)-form is{)[) ofreal-valued function). LetMn be a closed Kahler manifold. If w is an exact real (1, 1)-form, then there
exists a real-valued function'!/; such that Ao8'l/; = w. That is,
a2