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arXiv:math/0703486v1 [math.DG] 16 Mar 2007
K¨AHLER-RICCI FLOW ON A TORIC MANIFOLD
WITH POSITIVE FIRST CHERN CLASS
Xiaohua Zhu∗
Abstract. In this note, we prove that on an n-dimensional compact toric manifold with
positive first Chern class, the K¨ ahler-Ricci flow with any initial (S1)n-invariant K¨ ahler
metric converges to a K¨ ahler-Ricci soliton. In particular, we give another proof for the
existence of K¨ ahler-Ricci solitons on a compact toric manifold with positive first Chern
class by using the K¨ ahler-Ricci flow.
0. Introduction.
Let M be a compact toric manifold with positive first Chern class. Let T∼= (S1)n×Rn
be a maximal torus which acts on M and K0∼= (S1)nbe its maximal compact subgroup.
In this note we discuss a K¨ ahler-Ricci flow with a K0-invariant initial metric on M and
we shall prove
Main Theorem. On a compact toric manifold M with positive first Chern class, the
K¨ ahler-Ricci flow with any initial K0-invariant K¨ ahler metric converges to a K¨ ahler-Ricci
soliton. In particular, it shows that there exists a K¨ ahler-Ricci soliton on any compact
toric manifold with positive first Chern class.
The existence of K¨ ahler-Ricci solitons on a compact toric manifold with positive first
Chern class was proved in [WZ] by using the continuity method. The above theorem
gives another proof for the existence of K¨ ahler-Ricci solitons on such a complex manifold
by using the K¨ ahler-Ricci flow. We note that a more general convergence theorem of
K¨ ahler-Ricci flow on a compact complex manifold which admits a K¨ ahler-Ricci soliton
was recently obtained by Tian and the author in [TZ3]. In that paper the assumption of
the existence of a K¨ ahler-Ricci soliton plays a crucial role. In the case of K¨ ahler-Einstein
manifolds with positive first Chern class the same result was claimed by Perelman ([P2]).
In the present paper we do not need any assumption of the existence of K¨ ahler-Ricci
solitons or K¨ ahler-Einstein metrics and prove the the convergence of K¨ ahler-Ricci flow.
The Ricci flow was first introduced by R. Hamilton in 1982 ([Ha]). Recently G. Perelman
has made a major breakthrough in this area for three-dimensional manifolds ([P1]).
Our proof of the main theorem is to study certain complex Monge-Amp` ere flow instead
of K¨ ahler-Ricci flow. The flow of this type has been studied before by many people (cf.
1991 Mathematics Subject Classification. Primary: 53C25; Secondary: 32J15, 53C55, 58E11.
Key words and phrases. Toric manifold, the K¨ ahler-Ricci flow, K¨ ahler-Ricci solitons.
* Partially supported by NSF10425102 in China and a Huo Y-D fund
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[Ca], [CT1], [CT2]). Indeed, our proof used a deep estimate of Perelman ([P2], also see
[ST]). We combined Perelman’s estimate with estimates on solutions of complex Monge-
Amp` ere flow appeared in [TZ3] and used an argument for C0-estimate on certain real
Monge-Amp` ere equation studied in [WZ].
The organization of this paper is as follows: In Section 1, We describe an unpublished
estimate of Perelman on the time derivative of potential functions of evolved K¨ ahler
metrics along the K¨ ahler-Ricci flow. In Section 2, we reduce the K¨ ahler-Ricci flow to
a real Monge-Amp` ere flow in order to get an upper bound of solutions of potential
functions. Then in Section 3, we use an argument in [TZ3] to get a C0-estimate of
solution. The main Theorem will be proved in Section 4.
1. An estimate of Perelman.
In this section, we first reduce the K¨ ahler-Ricci flow to a fully nonlinear flow on K¨ aher
potentials. Then we discuss a recent and deep estimate of Perelman.
Let (M,g) be an n-dimensional compact K¨ ahler manifold with its K¨ ahler form ωg
representing the first Chern class c1(M) > 0. In local coordinates z1,··· ,zn, we have
√−1
2π
i,j=1
ωg=
n
?
gijdzi∧ dzj, gi¯j= g(∂
∂zi,
∂
∂zj).
Moreover, the Ricci form Ric(ωg) is given by
?
Rij= −∂i∂jlog(det(gkl)),
Ric(ωg) =
2π
√−1
?n
i,j=1Rijdzi∧ dzj.
Since the Ricci form represents c1(M), there exits a smooth function h on M such that
Ric(ωg) − ωg=
√−1
2π
∂∂h.
(1.1)
The Ricci flow was first introduced by R. Hamilton in [Ha]. If the underlying manifold
M is K¨ ahler with positive first Chern class, it is more natural to study the following
K¨ ahler-Ricci flow (normalized),
?
∂g(t,·)
∂t
g(0,·) = g0,
= −Ric(g(t,·))+ g(t,·),
(1.2)
where g0is a given metric with its K¨ ahler class representing c1(M). It can be shown that
(1.2) preserves the K¨ ahler class, so we may write the K¨ ahler form of g(t) at a solvable
time t as
ωϕ= ωg+
√−1
2π
∂∂ϕ
for some smooth function ϕ = ϕ(t,·) = ϕt. This ϕ is usually called a K¨ ahler potential
function associated to the K¨ ahler metric g(t). Using the Maximal Principle, one can
show that (1.2) is equivalent to the following complex Monge-Amp` ere flow for ϕ(t,·),
?
ϕ(0,·) = 0.
2
∂ϕ
∂t= log
det(gij+ϕij)
det(gij)
+ ϕ − h
(1.3)
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where h is normalized as in Proposition 3.2. Note that θ˜
3.2, we have
Xt=˜Xt(u0). So by Proposition
|∂ ˜ ϕ
∂t| = |∂ϕ
∂t+ θ˜
Xt| ≤ C
Proof of Main Theorem. We shall show that the corresponding K¨ ahler metrics ω˜ ϕasso-
ciated to solution of equation (4.4) converge to a K¨ ahler-Ricci soliton ωKSwith respect
to X. The proof is similar to one of Main theorem in [TZ3]. We give a sketch. First by
modifying Yau’s C2-estimate in [Ya] for certain complex Monge-Amp` ere equation, one
obtains for the solution ˜ ϕ of equation (4.4),
?˜ ϕ?C2(M)≤ C and (gij+ ˜ ϕij) > c0.
Then following Calabi’s C3-estimate for complex Monge-Amp` ere equation [Ya], we fur-
ther get
?˜ ϕ?C3(M)≤ C.
Thus by using regularity theory for the parabolic equation, one sees easily that all Ck-
norms of the solution ˜ ϕ are uniformly bounded. Therefore, we conclude that for any
sequence of functions ˜ ϕt, one can take a subsequence of the sequence which converge
Ck-smoothly to a smooth function ϕ∞on M.
Let σt= exp{tX} and σ′= σ′
corresponding to changes from ϕtto ˜ ϕt. Since
t= ρt· σ−1
t , where ρtare holomorphic transformations
?
M
?∂((σ′)∗∂ϕ′
∂t)?2eθX+X(˜ ϕ)ωn
˜ ϕ=
?
M
?∂∂ϕ′
∂t?2eθX+X(ϕ′)(ωϕ′)n,
then by (3.3), we have
?
M
?∂((σ′)∗∂ϕ′
∂t)?2eθX+X(˜ ϕ)ωn
˜ ϕ= −d˜ µωg(ϕ′)
dt
.
(4.5)
By the lower bound of ˜ µωg(ϕ′) (cf. Corollary 3.1), one sees that there is a sequence of
ti, i = 1,2,..., such that
?
M
?∂((σ′)∗∂ϕ′
∂t|ti)?2ωn
˜ ϕti→ 0, as i → ∞.
(4.6)
On the other hand, by (4.4), we have
√−1
2π
∂∂[(σ′)∗∂ϕ′
∂t] = −Ric(ω˜ ϕ) + ω˜ ϕ+ LXω˜ ϕ.
(4.7)
Then (σ′)∗ ∂
sequence of (σ′)∗ ∂
same indices ti) converge to a constant in the Cksense, and consequently, by (4.7),
K¨ ahler metrics ω˜ ϕticonverge to a K¨ ahler-Ricci soliton (ωg+
the holomorphic vector field X. It remains to prove that the limit (ωg+
is independent of the choice of sequence of ω˜ ϕt. But the last follows from the uniqueness
of K¨ ahler-Ricci solitons proved in [TZ1] and [TZ2]. We leave the details to reader.
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∂tϕ′is Ckuniformly bounded in space, so there exists a convergent sub-
∂tϕ′(ti,·). Hence by (4.6), we conclude that (σ′)∗ ∂
∂tϕ′(ti,·) (still use
√−1
2π∂∂ϕ∞) associated to
√−1
2π∂∂ϕ∞,X)
?
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Department of Mathematics, Peking University, Beijing, 100871, China
E-mail address: xhzhu@math.pku.edu.cn
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