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arXiv:math/0505476v3 [math.DG] 12 Jan 2009
ENERGY FUNCTIONALS AND CANONICAL K¨AHLER METRICS1
Jian SongBen Weinkove
Harvard University
Department of Mathematics
Cambridge MA 02138
Johns Hopkins University
Department of Mathematics
Baltimore MD 21218
Abstract.
if and only if it is stable in the sense of geometric invariant theory. There has been
much progress on this conjecture by Tian, Donaldson and others. The Mabuchi energy
functional plays a central role in these ideas. We study the Ekfunctionals introduced
by X.X. Chen and G. Tian which generalize the Mabuchi energy. We show that if a
Fano manifold admits a K¨ ahler-Einstein metric then the functional E1is bounded from
below, and, modulo holomorphic vector fields, is proper. This answers affirmatively a
question raised by Chen. We show in fact that E1is proper if and only if there exists a
K¨ ahler-Einstein metric, giving a new analytic criterion for the existence of this canonical
metric, with possible implications for the study of stability. We also show that on a
Fano K¨ ahler-Einstein manifold all of the functionals Ekare bounded below on the space
of metrics with nonnegative Ricci curvature.
Yau conjectured that a Fano manifold admits a K¨ ahler-Einstein metric
Mathematical Subject Classification: 32Q20 (Primary), 53C21 (Secondary)
1. Introduction
The problem of finding necessary and sufficient conditions for the existence of ex-
tremal metrics, which include K¨ ahler-Einstein metrics, on a compact K¨ ahler manifold
M has been the subject of intense study over the last few decades and is still largely
open. If M has zero or negative first Chern class then it is known by the work of
Yau [Ya1] and Yau, Aubin [Ya1], [Au] that M has a K¨ ahler-Einstein metric. When
c1(M) > 0, so that M is Fano, there is a well-known conjecture of Yau [Ya2] that
the manifold admits a K¨ ahler-Einstein metric if and only if it is stable in the sense of
geometric invariant theory.
There are now several different notions of stability for manifolds [Ti2], [PhSt1],
[Do2], [RoTh]. Donaldson showed that the existence of a constant scalar curvature
metric is sufficient for the manifold to be asymptotically Chow stable [Do1] (under
an assumption on the automorphism group). It is conjectured by Tian [Ti2] that the
existence of a K¨ ahler-Einstein metric should be equivalent to his ‘K-stability’. This
stability is defined in terms of the Futaki invariant [Fu], [DiTi] of the central fiber of
degenerations of the manifold. Donaldson [Do2] introduced a variant of K-stability
extending Tian’s definition.
1The second author is supported in part by National Science Foundation grant DMS-05-04285
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The behavior of Mabuchi’s [Mb] energy functional is central to this problem. It was
shown by Bando and Mabuchi [BaMb], [Ba] that if a Fano manifold admits a K¨ ahler-
Einstein metric then the Mabuchi energy is bounded below. Recently, it has been shown
by Chen and Tian [ChTi3] that if M admits an extremal metric in a given class then
the (modified) Mabuchi energy is bounded below in that class. Donaldson has given an
alternative proof for constant scalar curvature metrics with a condition on the space of
automorphisms [Do3]. Moreover, if a lower bound on the Mabuchi energy is given then
the class is K-semistable [Ti2], [PaTi]. Conversely, Donaldson [Do2] showed that, for
toric surfaces, K-stability implies the lower boundedness of the Mabuchi energy.
In addition, the existence of a K¨ ahler-Einstein metric on a Fano manifold has been
shown to be equivalent to the ‘properness’ of the Mabuchi energy [Ti2]. Tian conjectured
[Ti3] that the existence of a constant scalar curvature K¨ ahler metric be equivalent to this
condition on the Mabuchi energy. This holds when the first Chern class is a multiple
of the K¨ ahler class. (If c1(M) < 0, it has been shown [Ch1], [We], [SoWe] that the
Mabuchi energy is proper on certain classes which are not multiples of the canonical
class. It is not yet known whether there exists a constant scalar curvature metric in
such classes.)
In this paper, we discuss a family of functionals Ek, for k = 0,...,n, which were
introduced by Chen and Tian [ChTi1]. They are generalizations of the Mabuchi energy,
with E0 being precisely Mabuchi’s functional. The functionals Ek can be described
in terms of the Deligne pairing [De]. This construction of Deligne has provided a very
useful way to understand questions of stability [Zh], [PhSt1], [PhSt2]. Phong and Sturm
[PhSt3] show that, up to a normalization term, the Mabuchi energy corresponds to the
Deligne pairing ?L,...,L,K−1?. Generalizing this, the functionals Ekcan be described
in terms of the pairing:
n−k
?
The fact that the functionals Ekcan be formulated in this way seems now to be known by
some experts in the field, and was pointed out to us by Jacob Sturm in 2002. However,
since it does not appear in the literature, we have included a short explanation of this
correspondence (see section 2).
A critical metric ω of Ekis a solution of the equation
?
???
L,...,L,
k+1
??
?
?
K−1,...,K−1?.
σk+1(ω) − ∆(σk(ω)) = constant,
where σk(ω) is the kth elementary symmetric polynomial in the eigenvalues of the Ricci
tensor of the metric ω. Critical points for E0are precisely the constant scalar curvature
metrics. Critical points for Enare the metrics of constant central curvature as described
by Maschler [Ms]. K¨ ahler-Einstein metrics are solutions to the above equation for all
k. The critical metrics are discussed more in section 2.
The functionals Ekwere used by Chen and Tian [ChTi1, ChTi2] to obtain conver-
gence of the normalized K¨ ahler-Ricci flow on K¨ ahler-Einstein manifolds with positive
bisectional curvature (see [PhSt4] for a related result). The Mabuchi energy is decreas-
ing along the flow. The functional E1is also decreasing, as long as the sum of the Ricci
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curvature and the metric is nonnegative. A major part of the argument in [ChTi1] is
to show that the Ekcan be bounded from below along the K¨ ahler-Ricci flow assum-
ing nonnegative Ricci curvature along the flow and the existence of a K¨ ahler-Einstein
metric.
In a recent preprint, Chen [Ch2] has proved a stability result for E1for Fano mani-
folds in the sense of the K¨ ahler-Ricci flow2. In [Ch2], Chen asked whether E1is bounded
below or proper on the full space of potentials (not just along the flow) if there exists
a K¨ ahler-Einstein metric. In this paper we answer Chen’s question: E1 is bounded
below if there exists a K¨ ahler-Einstein metric, and the lower bound is attained by this
metric. Moreover, modulo holomorphic vector fields, E1is proper if and only if there
exists a K¨ ahler-Einstein metric. We also show that, again assuming the existence of a
K¨ ahler-Einstein metric, the functionals Ekare bounded below on the space of metrics
with nonnegative Ricci c urvature.
We now state these results more precisely. Let ω be a K¨ ahler form on the com-
pact manifold M of complex dimension n. Write P(M,ω) for the space of all smooth
functions φ on M such that
ωφ= ω +√−1∂∂φ > 0.
For φ in P(M,ω), let φtbe a path in P(M,ω) with φ0= 0 and φ1= φ. The functional
Ek,ωfor k = 0,...,n is defined by
?1
−n − k
V
0
M
where V is the volume?
?
Ek,ω(φ) =k + 1
V
0
?
?
M
(∆φt˙φt)Ric(ωφt)k∧ ωn−k
φt
dt
?1
˙φt(Ric(ωφt)k+1− µkωk+1
φt) ∧ ωn−k−1
φt
dt, (1.1)
Mωn, and µkis the constant, depending only on the classes [ω]
and c1(M) given by
µk=
MRic(ω)k+1∧ ωn−k−1
?
Mωn
= (2π)k+1[K−1]k+1· [ω]n−k−1
[ω]n
.
The functional is independent of the choice of path. We will often write Ek(ω,ωφ)
instead of Ek,ω(φ).
In fact, Chen and Tian [ChTi1] first define Ekby a different (and explicit) formula,
making use of a generalization of the Liouville energy, which they call E0
pairing provides another explicit formula (Proposition 2.1).
Suppose now that M has positive first Chern class and denote by K the space of
K¨ ahler metrics in 2πc1(M). Notice that for ω in K, the corresponding constant µkis
equal to 1. We have the following theorem on the lower boundedness of the functionals
Ek.
k. The Deligne
2Shortly after this paper was first posted we learned that, in an unpublished work [Ch3], Chen
has proved the following: if there exists a K¨ ahler-Einstein metric then E1 is bounded below along the
K¨ ahler-Ricci flow.
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Theorem 1.1 Let (M,ωKE) be a K¨ ahler-Einstein manifold with c1(M) > 0. Then, for
k = 0,...,n, and for all ˜ ω ∈ K with Ric(˜ ω) ≥ 0,
Ek(ωKE, ˜ ω) ≥ 0,
and equality is attained if and only if ˜ ω is a K¨ ahler-Einstein metric.
In the case of E1we obtain lower boundedness on the whole space K. In addition,
it is an easy result that for a Calabi-Yau manifold, E1is bounded below on every class.
Putting these two cases together we obtain:
Theorem 1.2 Let (M,ωKE) be a K¨ ahler-Einstein manifold with c1(M) > 0 or c1(M) =
0. Then for all K¨ ahler metrics ω′in the class [ωKE],
E1(ωKE,ω′) ≥ 0,
and equality is attained if and only if ω′is a K¨ ahler-Einstein metric.
We show that if (M,ωKE) is K¨ ahler-Einstein with c1(M) > 0 and if there are no
holomorphic vector fields, E1is bounded below by the Aubin-Yau energy functional J
raised to a small power. This implies that E1is proper on P(M,ωKE). If there exist
holomorphic vector fields, then the statement changes slightly (c.f. [Ti2]).
Theorem 1.3 Let (M,ωKE) be a compact K¨ ahler-Einstein manifold with c1(M) > 0.
Then there exists δ depending only on n such that the following hold:
(i) If M admits no nontrivial holomorphic vector fields then there exist positive con-
stants C and C′depending only on ωKEsuch that for all θ in P(M,ωKE),
E1,ωKE(θ) ≥ CJωKE(θ)δ− C′.
(ii) In general, let G be the maximal compact subgroup of Aut0(M) which fixes ωKE,
where Aut0(M) is the component of the automorphism group containing the iden-
tity. Then there exist positive constants C and C′depending only on ωKE such
that for all θ in PG(M,ωKE),
E1,ωKE(θ) ≥ CJωKE(θ)δ− C′,
where PG(M,ωKE) consists of the G-invariant elements of P(M,ωKE).
Remark 1.1 The analagous result above is proved in [Ti2], [TiZh] for the F functional
[Di], giving a generalized Moser-Trudinger inequality. We use a similar argument. The
corresponding inequality for the Mabuchi energy also holds [Ti3] since, up to a constant,
it can be bounded below by F. It would be interesting to find the best constant δ = δ(n)
for which these inequalities hold. With some work, modifying the argument in [Ti2],
one can show that δ can be taken to be arbitrarily close to 1/(4n + 1), but we doubt
that this is optimal.
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Proof of Theorem 1.4
We have to show that if E1 is proper then there exists a K¨ ahler-Einstein metric
on M. Fix a metric ω and consider again the family of Monge-Amp` ere equations
ωn
we have a solution for t = 1 then ω +√−1∂∂φ1is a K¨ ahler-Einstein metric. We use
the continuity method. We have a solution φ0 at t = 0 by Yau’s theorem [Ya1]. It
is well-known that we can find a solution φtfor 0 ≤ t ≤ 1 as long as we can bound
the C0norm of φtuniformly in t. Moreover, ?φt?C0 ≤˜C(1 + Jω(φt)) for a constant
˜C depending only on ω. So it suffices to show that Jω(φt) is uniformly bounded from
above. Since E1,ω is proper, it suffices to show that E1,ω(φt) is bounded from above
uniformly in t.
We apply Lemma 5.3 in the case when t2= t and t1= 0. Then
φt= e−tφt+fωn, where f is given by Ric(ω) − ω =
√−1∂∂f and
?
Mefωn= V. If
E1,ω(φt) − E1,ω(φ0) = −2(1 − t)(Iω− Jω)(φt) + 2(Iω− Jω)(φ0)
− 2
0
+(1 − t)2
V
−1
V
M
?t
(Iω− Jω)(φs)ds
?
√−1∂φ0∧ ∂φ0∧ ωn−1
?t
√−1∂φ0∧ ∂φ0∧ ωn−1
M
√−1∂φt∧ ∂φt∧ ωn−1
φt
?
φ0
≤ 2(Iω− Jω)(φ0) − 2
−1
V
M
0
(Iω− Jω)(φs)ds
?
φ0,
where we are using the definition of Iω− Jω. It then follows immediately that E1,ω(φt)
is bounded from above by a constant independent of t. Q.E.D.
Acknowledgements. The authors would like to thank: Professor D.H. Phong, for
his constant support and advice; Professor J. Sturm, for a number of very enlightening
discussions; Professor X.X. Chen, for some helpful suggestions and in particular for
bringing our attention to the functional E1; Professor S.-T. Yau, for his encouragement
and support; Professor S.T. Paul for some useful discussions; and the referees for some
helpful comments.
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