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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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Remark 1.2 We hope that the above results on the E1 functional may have some

applications to the study of the stability of M. Indeed, let π1: X → Z be an SL(N +

1,C)-equivariant holomorphic fibration between smooth varieties such that X ⊂ Z ×

CPNis a family of subvarieties of dimension n with an action of SL(N +1,C) on CPN.

Tian defines CM-stability [Ti2] for Xz= π−1

1(z) in terms of the virtual bundle:

E = (K−1− K) ⊗ (L − L−1)n−

nµ0

n + 1(L − L−1)n+1,

where K = KX⊗ K−1

hyperplane bundle on CPNvia the second projection π2(alternatively, one can use the

language of the Deligne pairing). When Xz is Fano, Tian proved that Xz is weakly

CM-stable if it is K¨ ahler-Einstein, using the properness of the Mabuchi energy E0. One

can define a similar notion of stability for Xzwith respect to the virtual bundle

Z

is the relative canonical bundle and L is the pullback of the

Ek= (K−1− K)k+1⊗ (L − L−1)n−k−(n − k)µk

For k = 1, one might guess that Xzis stable if it is K¨ ahler-Einstein, since E1is proper.

It would be interesting to try to relate this notion of stability to an analogue of K-

stability expressed in terms of the holomorphic invariants Fk[ChTi1] which generalize

the Futaki invariant.

n + 1

(L − L−1)n+1.

We also have a converse to Theorem 1.3.

Theorem 1.4 Let (M,ω) be a compact K¨ ahler manifold with c1(M) > 0. Suppose that

ω ∈ 2πc1(M). Then the following hold:

(i) Suppose that (M,ω) admits no nontrivial holomorphic vector fields.

admits a K¨ ahler-Einstein metric if and only if E1is proper on P(M,ω).

Then M

(ii) In general, suppose that ω is invariant under G, a maximal compact subgroup of

Aut0(M). Then M admits a G-invariant K¨ ahler-Einstein metric if and only if

E1is proper on PG(M,ω).

This gives a new analytic condition for a Fano manifold to admit a K¨ ahler-Einstein

metric. Indeed, together with the result of Tian [Ti2], at least modulo holomorphic

vector fields, we have:

M admits a

K¨ ahler-Einstein

metric

E1is proper⇐⇒⇐⇒ Mabuchi energy is proper

and one might expect some versions of stability to be equivalent to these as well.

Remark 1.3 It is natural to ask whether there exist critical metrics for Ekwhich are

not K¨ ahler-Einstein. Chen and Tian [ChTi1] observed that for k = n the only critical

metrics with positive Ricci curvature are K¨ ahler-Einstein. Maschler [Ms] proved this

result without the assumption on the Ricci curvature when c1(M) > 0 or c1(M) < 0.

We see from Theorem 1.2 that on a K¨ ahler-Einstein manifold with c1(M) > 0, a critical

metric for E1which is not K¨ ahler-Einstein could not give an absolute minimum of E1.

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